1. Adjust oven rack to middle position, place large rimmed baking sheet on rack, and heat oven to 500°F.

2. Cut one 12-inch sheet of foil and spread flat on counter. Place garlic heads, cut-side up, in center of foil. Drizzle 1/2 teaspoon oil over each head and seal packet. Place packet on oven rack and roast until garlic is very tender, about 40 minutes. Open packet and set aside to cool.

3. While garlic is roasting, prepare mushrooms. Remove and discard stems. Wipe caps clean and cut into 3/4-inch slices. Place mushrooms in large bowl; toss with 2 tablespoons oil, 1 teaspoon salt, pepper to taste, and sugar.

4. Remove baking sheet from oven. Carefully transfer mushrooms to baking sheet and spread into even layer. Return baking sheet to oven and roast until mushrooms are well browned and tender, 20 to 25 minutes, flipping mushrooms halfway through cooking.

5. While mushrooms roast, bring 4 quarts water to boil in large pot. Add 1 tablespoon salt and pasta; cook until al dente. Squeeze roasted garlic cloves from their skins into small bowl. Using fork, mash garlic to smooth paste, then stir in red pepper flakes and 2 tablespoons lemon juice. Slowly whisk in remaining 1/4 cup oil.

6. Drain pasta, reserving 1 cup cooking water, and return pasta to pot. Add chopped mushrooms to pasta; stir in garlic sauce, 1/4 cup cooking water, rosemary, and 1/2 cup cheese. Adjust consistency with additional cooking water and season with salt, pepper, and additional lemon juice to taste. Serve immediately, sprinkling with remaining 1/2 cup cheese and toasted pine nuts.

Serves 4