Adjust an 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. Place the packet directly on the oven rack and roast until the garlic is very tender, about 40 minutes.

Cut the cauliflower head from pole to pole into 8 equal wedges. Toss the wedges with oil, salt, pepper, and sugar in a large bowl. Carefully remove the hot baking sheet from the oven and lay the cauliflower, cut side down, in an even layer. Roast until the cauliflower is well browned and tender, 20 to 25 minutes.

Transfer the cauliflower to a cutting board, let it cool slightly, then cut into bite-sized pieces. Unwrap the garlic and, once cool enough to handle, squeeze the cloves from their skins into a small bowl. Mash the garlic smooth with a fork, stir in the lemon juice and pepper flakes, then slowly whisk in the oil until 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 a cup of the cooking water, then drain the pasta and return it to the pot.

Add the chopped cauliflower, garlic sauce, Parmesan, and parsley to the pot along with a splash of the reserved cooking water. Toss to combine, adding more water as needed until the sauce coats the pasta. Season with salt, pepper, and extra lemon juice to taste. Serve immediately, topped with walnuts and extra Parmesan.