Yea, I don't mind posting my version. I played with it some. Didn't make the GUI prettier, because I hate tweaking GUI's, but I did try to streamline the code a little.

I had to make a CardPlayer class to handle a blackjack player's hand contents (Q, K, 8, etc) and hand sum. This allows you to more easily expand this code to have more than 1 player. (Currently the Blackjack class cannot handle more than Dealer vs one Player, however).

I also had to change the algorithm for summing up a player's hand. The way you were doing it wasn't working consistently. My way eliminated the "values" dictionary because it was just too much hassle to check values repeatedly.

import random
from tkinter import *
deck = ["2", "3", "4", "5", "6", "7", "8", "9", "T", "J", "Q", "K", "A"]

class CardPlayer:
    global deck
    def __init__(self):
        self.hand = []
        self.hand_sum = 0

    def draw_card(self):
        card = random.choice(deck)
        self.hand.append(card)
        self.hand.sort()
        self._update_hand_value()

    def _update_hand_value(self):
        value =0
        aces = 0
        for card in self.hand:
            if str.isnumeric(card):
                value += int(card)
            else:
                if card == "A":
                    aces += 1
                    continue
                value += 10
        for ace in range(aces):
            if (value + 11)>21:
                value += 1
            else:
                value += 11

        self.hand_sum = value

class Blackjack:
    def __init__(self):
        self.__game = Tk()
        self.__game.geometry("700x300")

        # background
        #self.__table = PhotoImage(file=".....")
        self.__table= Label(self.__game)
        #self.__table.config(image = self.__table)
        self.__table.place(x=0,y=0, relwidth=1, relheight=1)

        # How to end the game
        self.__close_instructions = Label(self.__game, text=
            "Press 'q' to quit or 'a' to start a new game after this one ends")
        self.__close_instructions.grid(row=0, column=0, sticky=W)
        self.__game.config(cursor="tcross")
        self.__game.grid_rowconfigure(0, minsize=100)
        self.__game.grid_columnconfigure(0, minsize = 100)

        # Pressing q:closes window
        self.__game.bind("q", lambda q: self.__game.destroy())

        # Print your hand
        self.__post_cards = Label(self.__game)
        self.__post_cards.grid(row=2, column=1)

        # Print dealer’s cards
        self.__dealer_cards = Label(self.__game)
        self.__dealer_cards.grid(row=2, column=2)

        # Print your score
        self.__player_score = Label(self.__game)
        self.__player_score.grid(row=3, column=1)

        # Prints for each game
        self.__main_display = Label(self.__game)
        self.__main_display.grid()
        self.__main_display.config(text="Let's play Blackjack!")

        self.__main_display.config(text="Welcome to the table!")
        self.__start_button = Button(self.__game)
        self.__start_button.config(text = "Deal", command=self.begin_game)
        self.__start_button.grid()

        self.__game.mainloop()

    def begin_game(self):
        self.__start_button.config(state=DISABLED)
        self.__game.unbind("a") #unbind 'a' press so player must play

        # dealer draws
        self.Dealer = CardPlayer()
        self.Dealer.draw_card()
        self.Dealer.draw_card()

        # the player draws
        self.Player1 = CardPlayer()
        self.Player1.draw_card()
        self.Player1.draw_card()

        self.__post_cards.config(text="Your cards: %s" % " ".join(self.Player1.hand))
        self.__player_score.config(text="Your score: %d" % self.Player1.hand_sum)
        self.__dealer_cards.config(text="The dealer shows: %s " % (self.Dealer.hand[0]))
        self.__main_display.config(text="hit or stay?")

        # This button chooses hit
        self.__hitbutton = Button(self.__game)
        self.__hitbutton.config(text="Hit", command=lambda : self.hit())
        self.__hitbutton.grid(row=3, column=2)

        # This button chooses stay
        self.__staybutton = Button(self.__game)
        self.__staybutton.config(text="Stay", command=lambda : self.stay())
        self.__staybutton.grid(row=3, column=3)

        if self.Player1.hand_sum == 21:
            self.__main_display.config(text="BLACKJACK!!")
            self.__hitbutton.destroy()
            self.__staybutton.destroy()
            self.play_again()

    def hit(self):
        self.Player1.draw_card()
        self.__post_cards.config(text="Your cards: %s" % " ".join(self.Player1.hand))
        self.__player_score.config(text="Your score: %d" % self.Player1.hand_sum)

        if self.Player1.hand_sum==21:
            self.__main_display.config(text="Great Job!")
            self.__main_display.config(text="Let's see what the Dealer draws...")
            self.__hitbutton.destroy()
            self.__staybutton.destroy()
            self.dealer_hit()

        elif self.Player1.hand_sum>21:
            self.__main_display.config(text="Bust!! You lost this hand.")
            self.__hitbutton.destroy()
            self.__staybutton.destroy()
            self.play_again()

    def stay(self):
        self.__hitbutton.destroy()
        self.__staybutton.destroy()
        self.dealer_hit()


    def dealer_hit(self):
        self.__dealer_cards.config(
            text="The dealer shows: %s which totals %d" % (" ".join(self.Dealer.hand), self.Dealer.hand_sum))

        while self.Dealer.hand_sum<16:
            if self.Dealer.hand_sum<22 and self.Dealer.hand_sum<16:
                self.__main_display.config(text="Dealer must draw...")
            self.Dealer.draw_card()
            self.__dealer_cards.config(text="The dealer has: %s which totals %d" % (" ".join(self.Dealer.hand), self.Dealer.hand_sum))

        if self.Dealer.hand_sum>21:
            self.__main_display.config(text="Dealer BUSTS!! You Win!")

        elif self.Dealer.hand_sum<self.Player1.hand_sum:
            self.__main_display.config(text="You Win!!")

        elif self.Dealer.hand_sum==self.Player1.hand_sum:
            self.__main_display.config(text="PUSH, you both keep your bets.")

        elif self.Dealer.hand_sum>self.Player1.hand_sum:
            self.__main_display.config(text="Dealer Wins!!")

        self.play_again()  #enable 'a' press to start new game

    def play_again(self):
        self.__game.bind("a", lambda a: self.begin_game())

def main():
    blackjack = Blackjack()

main()