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Heat 1 tablespoon butter and 1 tablespoon olive oil in a wide-bottomed saucepan over medium-high heat. Add half the cremini mushrooms in a single layer and cook for 3 minutes without stirring to brown evenly. Stir and cook for 2 minutes more.
Transfer the mushrooms to a bowl. Add another tablespoon butter and olive oil and repeat with the remaining cremini mushrooms.
Repeat with the final 1/2 tablespoons butter, oil, and the shiitake mushrooms.
Return all the mushrooms to the pan and add the onion, carrot, celery, and garlic. Sprinkle with the kosher salt and stir to combine. Cook for 10 minutes, until the vegetables are very soft and browned bits are forming on the bottom of the pan.
Stir in the tomato paste and cook for 1 minute, then stir in the red wine and scrape up all the browned bits on the bottom of the pan.
Add the diced tomatoes and vegetable stock, and reduce the heat to medium-low. Simmer for 45 minutes, until the sauce is thick and deep crimson in color. Stir in the cream.
While the sauce simmers, bring 4 quarts of salted water to a boil. Add the fettuccine and cook according to package directions until al dente.
Drain the pasta and toss with the sauce. Serve with grated cheese if desired.