Adjust an oven rack to the middle position and heat to 400 degrees. Place the garlic heads on a small piece of aluminum foil, drizzle the exposed cloves with a teaspoon of oil, and wrap the foil tightly around the heads.

Roast the garlic directly on the oven rack until the cloves are very soft and golden brown, about 40 to 45 minutes.

Toss the broccoli florets with three tablespoons of oil, sugar, salt, and pepper on a rimmed baking sheet. Roast on the rack below the garlic until the florets are tender and the edges are deeply browned, 15 to 20 minutes, tossing halfway through for even color.

Bring a large pot of salted water to a boil and cook the pasta until al dente. Reserve at least one cup of the pasta cooking water before draining the noodles well.

Squeeze the softened roasted garlic cloves from their skins into a large, warm serving bowl. Use a fork to mash the garlic into a smooth paste with the remaining olive oil, lemon juice, and red pepper flakes.

Add the drained pasta, roasted broccoli, Manchego, basil, and almonds to the garlic paste. Toss vigorously to coat, adding the reserved pasta water a few tablespoons at a time until the cheese melts and a light, glossy sauce forms. Serve immediately with extra cheese and a squeeze of lemon if needed.