My go-to spaghetti sauce is almost vegetarian, and can be made so with one omission or substitution.

1 white onion, diced
2-3 medium carrots, diced
3-4 cloves garlic, sliced
2 28-oz cans whole plum tomatoes or equivalent in good fresh tomatoes, drained (buy the variety with the least salt so you can add salt yourself later)
1 glass decent white wine (I usually use pinot grigio or sauvignon blanc) 1 tbsp anchovy fillets, minced 1 tbsp fresh basil, chiffonaded 1 tbsp fresh oregano, minced

Wet the bottom of a medium-sized saute pan (I use this 12″, 5-quart one specifically) with olive oil and sweat the onion over medium heat with salt until translucent, about 8 minutes. Add the garlic and cook another 2-3 minutes. Add the carrot, with more salt, and continue to cook slowly another 12 minutes, being mindful that the garlic does not burn.

Add the tomatoes (having taking care to drain them of excess juice) and additional salt and continue to cook another 12 minutes, allowing a fond to form on the bottom of the pan. Add the wine and deglaze. Turn the flame to high to bring to a boil just for an instant, then reduce flame to its lowest setting to simmer.

Add anchovy, basil and oregano. At this point the sauce should still taste very raw, of uncooked wine and raw tomatoes, but taste for seasoning. If you're using fresh tomatoes, first of all congratulations, because unless you have a garden or live near farms it's damn near impossible to get fresh tomatoes that taste like anything, but more importantly you're going to need to add considerable salt, on the order of one to two teaspoons at least. If using canned tomatoes as is more likely, the salt you need to add will depend on how much was added to the tomatoes when they were canned. So as always, taste, season and then taste again.

Simmer on the stovetop for at least one hour. You'll know it's ready when it starts smelling and tasting like spaghetti sauce rather than like raw tomatoes. As it simmers the sauce will thicken. If it becomes too thick, you can add additional wine, bring back to a boil then let it simmer some more, or (my preference) you can add a little water from the pot you're cooking your spaghetti in. That water has a lot of dissolved starch in it, so it works well to loosen the sauce without making it unpleasantly watery.

Now, a vegetarian would understandably be put off by the inclusion of anchovies. They're in the recipe for their glutamic acid content, which is hugemongous. Glutamic acid is an amino acid that adds a very intense and very delicious flavor to savory dishes. Tomatoes have some glutamic acid in them, but having made this recipe many times both with and without the anchovies, my opinion (and a very strongly held one at that) is that the extra glutamic acid makes a huge difference.

So if you leave them out, you'll end up with what I think is an inferior sauce … which means we need to talk substitutions. Worcestershire sauce is an obvious alternative, but since it's actually made from anchovies, I guess it would be no more acceptable to a vegetarian than the anchovies themselves.

So what I'd try first would be mushrooms, specifically dried wild mushrooms. The porcini is the king of mushrooms as far as I'm concerned, but what we're looking for here isn't the flavor of mushroom but rather just the glutamic acid that mushrooms will contribute as they simmer in the sauce. So I'd recommend taking a small amount of dried mushrooms and mincing them, rehydrating them in wine overnight in the fridge, and then adding the wine to the sauce as normal. The dissolved glutamate will (I think) bring that richness of flavor we want. You can discard the mushrooms themselves; after a long soak in wine, they'll be quite flavorless and their texture won't be worth much either.

Anyway, that's what I'd try. I've never actually done that myself, but it'd be where I'd start.