Preheat the oven to 350°F (175°C).
Bring a large pot of salted water to a boil. Add the pasta and cook until just barely al dente, about 8 minutes or roughly two-thirds of the package time. Drain and set aside.
In a large skillet over medium heat, warm the olive oil. Add the sliced mushrooms and sauté until softened, about 6–7 minutes.
Add the tomato-based pasta sauce to the skillet with the mushrooms, bring to a boil, then reduce heat to low and simmer about 15 minutes.
In an 8 x 8-inch baking dish, layer half of the cooked pasta, then spread half of the tomato-mushroom sauce over it.
Arrange the provolone slices over the sauce. Dollop the sour cream in spoonfuls and gently spread into a thin layer, then sprinkle the crumbled Gorgonzola evenly on top.
Add the remaining pasta, then top with the shredded mozzarella. Spread the remaining tomato sauce over the top and finish by sprinkling the grated Parmesan.
Bake in the preheated oven for about 30 minutes, until the cheeses are bubbly and the top is lightly golden. Remove and let rest a few minutes before serving.