Recipe for the buns is easy:

380 grams AP flour

50 grams bread flour

1/2 tbs salt

2 tbs sugar

8 ounces warm water

2 ounces warm whole milk

1 packet instant dry yeast

3 tbs unsalted butter

1 egg

Beat the egg in with the milk, add the water and the sugar and yeast and stir until all dissolved.

Mix the flour and the salt. Mix the liquid into the dry ingredients with a wooden spoon until incorporated, then knead. I used a dough hook, and I kneaded it for about 6 minutes. While it's kneading, add the butter in 1 tbs at a time, allowing each tbs to knead in well before adding the next. Put in an oiled bowl in a warmish place and let rise until doubled (1-2 hours but mine only took 70 minutes). Punch down, cut into 8 equal portions, shape on a nicely floured surface, and then place them on a baking sheet and let them sit another hour (or more, depending on the temperature of your house). Brush with an egg wash and then bake at 400F for 14-16 minutes.

For the burgers, I shaped them ahead of time but didn't salt them until right before cooking. I cooked them to a medium and let them rest while I assembled everything. Really these were very easy to make and very tasty!