Place ½ of the mushrooms in a food processor, and pulse several times until finely ground. (Don't over process - see notes.) Transfer the ground mushrooms to a bowl, then add the rest of the mushrooms and repeat.
Place the walnuts in the food processor and process until well ground. Add them to the ground mushrooms.
Heat 1 tablespoon of water in a large pot or Dutch oven over medium-high heat. Add the mushrooms and the walnuts and cook until the all of the liquid has evaporated and the mushrooms have browned, about 8-10 minutes. Remove the mushroom/walnut mixture from the pot and set aside. (Don't clean out the pan; if there are areas of burned food, just scrape those out.)
Make a soffritto. Wipe out the food processor, and add the onion, carrots, celery and garlic. Process into a rough 'pulp,' about 30 seconds.
Heat 1 tablespoon of water in the pot or Dutch oven over medium-high heat. Add the soffritto (onions, carrots, celery and garlic) with a pinch of salt to the pot. Saute until all of the liquid cooks off, the color turns from bright orange to a deeper orange-brown, and the volume is reduced by about half, about 10-12 minutes. Use the liquid released from the pulp to deglaze (loosen) any dark brown remnants of the mushroom/walnut mixture and blend it in. Stir and scrape loose any browned bits from the pan if they form. Add water a little at a time if things begin to stick or get too brown.
Move the browned pulp aside to the edges of the pot, leaving an empty spot in the center. Add the tomato paste to the center of the pot, and cook about 5 minutes until it begins to brown and get dark, but not burn. (Keep stirring and scraping, adding water 1 teaspoon at a time and reducing heat a bit only if things start to get too dark or burn.) When the tomato paste has browned, stir it into the vegetable pulp.
Add the wine (or vinegar and 1 cup of water) to the pot to deglaze (loosen) and scrape up all of the brown bits ("fond") that have formed in the pan--this adds flavor!
Add the mushroom/walnut mixture, and stir well to combine everything.
Add the smashed tomatoes and their liquid, water, and crushed red pepper flakes to the pot and stir well. Bring the sauce to a boil, lower the heat, and simmer, uncovered for 5 minutes so that the alcohol from the wine cooks off.
Add the thyme bundle and bay leaf, cover, and simmer for at least 30 minutes--or even better, one hour or more.
Before serving, remove the thyme bundle (and any loose stems) and the bay leaf. Season with salt and pepper to taste.
Serve with pasta or polenta.
Notes
See the "pro tips" in the article above for more details to make this recipe perfect every time.Brown the soffritto and tomato paste very well. The more you brown these ingredients, the richer your sauce will taste.Simmer long and low if possible. The minimum simmer time for this recipe is 30 minutes, but an hour or longer will produce a richer sauce.Don't rinse mushrooms under water. Cleaning mushrooms is a bit of a pain, so you might be tempted to wash them in water. DON'T! They won't brown as well. Don't over process the mushrooms. For the most appealing texture, you don't wan't them to get too mealy or to turn into a paste. See the article pics as a guide.Tip for dicing whole canned tomatoes without a mess. Pour them tomatoes into a large bowl, and mash them up with a potato masher. Use kitchen shears to cut up any large pieces.Time saving tip for thyme. Tie the herbs into a bundle with kitchen string. But do it very tightly, or you'll be picking stems our of your sauce.