Adjust the oven rack to the middle position, place a large rimmed baking sheet on the rack, and heat the oven to 500°F. Drizzle the halved garlic heads with a little oil, wrap them securely in a foil packet, and roast on the oven rack until the cloves are very tender, about 40 minutes.

Toss the cauliflower wedges with oil, salt, pepper, and sugar in a large bowl until evenly coated. Carefully lay the cauliflower cut-side down on the hot baking sheet in the oven and roast until the edges are well-browned and the centers are tender, 20 to 25 minutes.

Transfer the cauliflower to a cutting board and chop into bite-sized pieces. Squeeze the roasted garlic cloves from their skins into a small bowl, mash them into a smooth paste with a fork, then whisk in the lemon juice, red pepper flakes, and the remaining olive oil until emulsified.

Bring a large pot of water to a boil, add the salt and pasta, and cook until al dente. Reserve 1 cup of the starchy cooking water before draining the pasta, then return the noodles to the pot.

Add the chopped cauliflower, garlic sauce, Parmesan, and parsley to the pot with the pasta, along with a splash of the reserved cooking water. Toss vigorously to combine, adding more water as needed until the sauce is glossy and coats every noodle. Season with salt, pepper, and extra lemon juice to taste, then serve topped with walnuts and extra cheese.