Pulse the cremini mushrooms in a food processor in batches until the pieces are no larger than ½ inch, then transfer to a large bowl.

Pulse the carrot and onion in the empty processor until finely chopped and add them to the bowl with the mushrooms.

Pulse the tomatoes and their juice in the empty processor until chopped fine and set aside in a separate bowl.

Melt the butter in a Dutch oven over medium heat. Add the mushroom mixture and dried porcini, cover, and cook until the vegetables release their liquid, about 5 minutes.

Uncover the pot and increase the heat to medium-high. Cook until the liquid has evaporated and the vegetables begin to brown deeply, 12 to 15 minutes.

Stir in the garlic and sugar and cook until fragrant, then add the tomato paste and cook for 1 minute to mellow its acidity.

Pour in the wine and simmer, scraping up any browned bits from the bottom of the pot, until the liquid is nearly evaporated, about 5 minutes.

Stir in the processed tomatoes, vegetable broth, soy sauce, salt, and pepper and bring to a simmer.

Reduce the heat to medium-low and simmer until the sauce has thickened but remains moist, 8 to 10 minutes.

While the sauce simmers, boil the pasta in a large pot of salted water until al dente. Reserve a small amount of cooking water before draining the pasta.

Remove the sauce from the heat and stir in the cream.

Toss the drained pasta with the sauce, adding the reserved cooking water as needed to reach a glossy consistency. Serve immediately with Parmesan.