Position an oven rack in the middle, place a rimmed baking sheet on it, and heat the oven to 500°F. Drizzle the halved garlic heads with oil and wrap them tightly in aluminum foil, then roast on the oven rack until the cloves are soft and golden, about 40 minutes.

Toss the mushrooms with oil, salt, pepper, and sugar in a large bowl. Carefully lay the mushrooms gill-side down onto the preheated baking sheet and roast until they are well-browned and tender, 20 to 25 minutes, flipping them halfway through to ensure even coloring.

Transfer the mushrooms to a cutting board and cut into half-inch pieces once cooled. Squeeze the roasted garlic cloves from their skins into a bowl and mash into a smooth paste with a fork, then whisk in the lemon juice and red pepper flakes before slowly adding the remaining oil until emulsified.

Boil the pasta in a large pot of water with the salt until al dente. Reserve a cup of the starchy cooking water before draining the pasta and returning it to the pot.

Combine the pasta with the chopped mushrooms, garlic sauce, cheese, and rosemary. Toss everything together, adding the reserved water a little at a time until the sauce is glossy and coats the pasta evenly. Serve immediately, topped with pine nuts and extra cheese.