Roast the cauliflower: Preheat the oven to 450 degrees F. Remove the green leaves from the cauliflower and reserve. Cut the cauliflower heads in half, then into bite-size florets. Cut the stalks into bite-size pieces also. Place the florets and stalk pieces in a roasting pan, and drizzle with olive oil and season with salt and pepper. Roast until golden and tender, about 20 minutes. Transfer to a platter lined with paper towels to drain.

Make the garlic chips: Pour 1/4 inch of oil into a small saucepan. Add the garlic and set the pan over medium heat. Heat the oil until the chips are golden and crisp, about 1 minute. Drain on paper towels and set aside.

Make the anchovy sauce: Set a large skillet over medium-high heat. Add the olive oil and fry the anchovies, garlic, capers, and thyme, stirring with a wooden spoon to break up the anchovies and infuse the oil. Cook for 3 to 4 minutes, until fragrant. Add the cream and Parmesan, and bring to a simmer. Just before serving, add the roasted cauliflower pieces and fold in the reserved leaves. Finish with a squeeze of lemon juice, and top with the fried garlic chips, cracked black pepper, and shaved Parmesan.