Preheat the oven to 450°F. Place the garlic heads on a small piece of aluminum foil and drizzle with the teaspoon of oil. Wrap the garlic tightly in the foil and roast on the oven rack until the cloves are very soft and tender, about 30 minutes.

While the garlic is roasting, place a rimmed baking sheet on the middle rack of the oven to preheat. In a large bowl, toss the cauliflower wedges with 3 tablespoons of the oil, the sugar, 3/4 teaspoon of salt, and 1/4 teaspoon of pepper until evenly coated.

Carefully remove the hot baking sheet from the oven and spread the cauliflower in an even layer. Roast until the bottoms of the cauliflower are deeply browned and the florets are tender, 20 to 25 minutes.

Bring a large pot of salted water to a boil and cook the pasta until al dente. Reserve 1/2 cup of the cooking water, then drain the pasta and return it to the pot.

Squeeze the softened roasted garlic cloves from their skins into a large bowl. Add the remaining 3 tablespoons of oil, lemon juice, and red pepper flakes; mash with a fork until a smooth paste forms.

Add the hot pasta, roasted cauliflower, garlic sauce, Parmesan, and parsley to the pot. Toss gently to combine, adding the reserved pasta water as needed to create a light sauce that coats the noodles. Season with extra lemon juice, salt, and pepper to taste.

Serve immediately in warm bowls, topped with the toasted walnuts and extra Parmesan.