Place the dried mushrooms in a medium bowl, cover with the hot water, and let sit until reconstituted and soft, about 15 minutes. Drain the mushrooms and their liquid through a fine mesh strainer into a clean bowl, squeezing the mushrooms to extract as much liquid as possible. Reserve the liquid and roughly chop the mushrooms. Set aside.

Bring a large pot of salted water to a boil. Add the fettuccini and cook until al dente, 7 to 8 minutes for dried pasta.

While the pasta is cooking, heat the oil and melt the butter in a large skillet over medium heat. Add the shallots and cook, stirring, until soft and fragrant, about 2 minutes. Add the garlic and cook, stirring, until fragrant, 30 seconds. Add the chopped mushrooms, and cook, stirring, for 2 minutes. Add the reserved mushroom liquid, bring to a boil, and cook, stirring occasionally, until the liquid is nearly all evaporated, 1 to 1 1/2 minutes. Add the cream, thyme, salt, and pepper and return to a simmer. Cook, stirring occasionally, until the cream is reduced and thick, 4 to 5 minutes. Add the parsley and stir to incorporate.

Drain the pasta and add to the pan with the sauce, tossing well to coat. Add 4 tablespoons of the cheese, toss, and remove from the heat.

Divide the pasta between two large bowls, garnish each with 1 tablespoon of the remaining cheese, and serve immediately.