Adjust the oven rack to the middle position, place a large rimmed baking sheet on the rack, and heat the oven to 500 degrees. Place the garlic heads, cut side up, on a square of aluminum foil, drizzle with oil, and wrap securely before roasting on the oven rack until the cloves are very tender, about 40 minutes.

Cut the broccoli crowns from the stems; peel the tough outer layer from the stems and cut into 1/2-inch thick batons. Toss the stems and crown wedges with oil, salt, pepper, and sugar in a large bowl.

Carefully lay the broccoli cut-side down on the preheated baking sheet in an even layer. Roast until the broccoli is well-browned and tender, 10 to 15 minutes.

Transfer the roasted broccoli to a cutting board and chop into bite-sized pieces. Let the garlic packet rest for 10 minutes before unwrapping, then squeeze the cloves into a small bowl and mash smooth with a fork. Whisk in the lemon juice, red pepper flakes, and the remaining oil.

Boil the pasta with salt in a large pot until al dente. Reserve 1 cup of the cooking water, then drain the pasta and return it to the pot.

Toss the pasta with the broccoli, garlic sauce, Manchego, and basil. Stir in a splash of the reserved cooking water to reach a loose, silky consistency that coats the pasta. Serve immediately with a sprinkle of almonds and extra cheese.