Adjust the oven rack to the middle position, place a large rimmed baking sheet on the rack, and heat the oven to 500°F. Place the garlic heads, cut side up, in the center of 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.

Prepare the broccoli by peeling the tough outer layer from the stems and cutting them into 1/2-inch thick pieces, then cutting the crowns into wedges. Toss the broccoli with oil, salt, pepper, and sugar in a large bowl. Carefully spread the broccoli in an even layer on the hot baking sheet and roast until the edges are well browned and the stems are tender, 10 to 15 minutes.

Transfer the roasted broccoli to a cutting board and chop into half-inch pieces once cool enough to handle. Unwrap the garlic and squeeze the cloves from their skins into a small bowl, mashing them into a smooth paste with a fork. Whisk in the lemon juice and red pepper flakes, then slowly whisk in the remaining oil until the mixture is emulsified.

Bring 4 quarts of water to a boil in a large pot. Add the pasta and salt and cook, stirring often, until al dente. Reserve 1 cup of the starchy cooking water, then drain the pasta and return it to the pot.

Add the chopped broccoli, garlic sauce, Manchego, basil, and a splash of the reserved cooking water to the pasta. Toss vigorously to melt the cheese and combine, adding more water as needed until a creamy consistency is achieved. Serve immediately with the toasted almonds and extra cheese sprinkled over the top.