Bring a large pot of salted water to a boil. Add fusilli, cook 5 minutes, and drain. Drizzle with olive oil to prevent pasta from sticking together.

Bring 1 cup water to a boil. Place mushrooms in a bowl and pour boiling water over to rehydrate. Let stand until softened, about 20 minutes. Remove mushrooms with a slotted spoon and place on cutting board. Reserve ⅓ cup of mushroom liquid, avoiding any grit at the bottom of the bowl. Discard remaining liquid and roughly chop mushrooms.

Heat oven to 375°F. Melt 4 tablespoons butter in a medium saucepan over medium heat. Add flour and cook, whisking constantly, until fragrant and toasted, about 3 minutes. Slowly whisk in milk and cream. Stir in salt and cook, stirring constantly, until mixture begins to thicken, 3 to 5 minutes. Remove mixture from heat and stir in reserved mushroom liquid. Add mozzarella and provolone and let stand 5 minutes. Briefly stir to combine—some cheese may not melt, and that’s fine.

Toss pasta with cheese sauce, mushrooms, basil, and sage. Transfer to a 2-quart baking dish and dot with remaining butter. Cover with foil and bake 20 minutes. Remove foil and bake, uncovered, until golden and bubbling, about 10 minutes. Serve warm.