What to buy: Dried porcini mushrooms, which can be found at most supermarkets, may be pricey. But after you taste the meaty flavor of the vegetarian broth they produce, you’ll understand why we think they’re worth every penny.
Place mushrooms in a medium bowl and rinse with cold water, shaking them in the water once or twice. Using a slotted spoon, transfer mushrooms to a small pot, leaving any grit at the bottom of the bowl behind, and add enough water to cover, about 1 1/2 cups. Set over low heat and bring to a simmer (this can take up to 45 minutes). Remove from heat and immediately lift mushrooms out of the liquid using a slotted spoon. Chop mushrooms and set aside. Strain mushroom broth through a coffee filter or cheesecloth into a medium bowl. Decant broth again by pouring it into another bowl and leaving any grit at the bottom behind. Set broth aside.
Place parsley, garlic, and onion in a food processor and pulse into a coarse chop, about 10 pulses. Set aside.
Place a heavy 4-quart pot fitted with a lid over medium-low heat. Add oil and butter. When butter foams, add onion-parsley mixture. Cook until onions are translucent, about 6 minutes. Stir in chopped mushrooms, then stir in tomato paste until incorporated. Slowly stir in reserved mushroom broth. Fill one of the tomato paste cans with cold water and add it to the mixture. Repeat 4 or 5 more times (for a total of about 2 cups of water), stirring with each addition. Season with a big pinch each of salt and freshly ground black pepper. Cover and simmer over the lowest possible heat for 60 to 75 minutes, adding more liquid if the sauce becomes too thick and stirring from time to time. Taste and season again with salt and pepper as needed.