Preheat the oven to 450°F. Place the garlic heads on a square of aluminum foil, drizzle with a bit of the oil, and wrap the foil tightly to create a packet.

Toss the sliced mushrooms with a portion of the olive oil, the sugar, salt, and pepper on a rimmed baking sheet. Arrange the mushrooms in a single layer and place the garlic packet on the corner of the sheet.

Roast until the mushrooms are browned and tender and the garlic cloves are completely soft and spreadable, about 25 to 30 minutes.

Boil the fusilli in a large pot of salted water until al dente. Reserve about a cup of the cooking water before draining the pasta.

Squeeze the roasted garlic cloves from their skins into a large bowl and mash them into a smooth paste. Whisk in the remaining olive oil, lemon juice, red pepper flakes, and rosemary until the mixture is emulsified.

Add the pasta, mushrooms, pine nuts, and Pecorino to the bowl with the garlic mixture. Toss thoroughly, adding the reserved pasta water a splash at a time until the sauce glides over the noodles and the cheese creates a light, creamy coating. Serve immediately while still hot.