Preheat an oven to 400°F and line a baking sheet with parchment paper. Bring a large pot of salted water to a boil. Start the sauce below after the water has come to a boil.
Quarter and core the cauliflower. Discard the core. Cut cauliflower into 1/4-inch to ½-inch thick slices, and place on the prepared baking sheet along with garlic cloves. Drizzle cauliflower with olive oil, 2 pinches of kosher salt, and several turns of black pepper, tossing to combine. Roast until tender and golden in spots (about 20 minutes), flipping the cauliflower after 10 minutes. Season to taste with additional kosher salt. Set aside.
Warm olive oil in a large skillet over medium heat. Add the garlic, anchovy paste and red chile flakes, cooking just until the garlic turns fragrant (1 - 2 minutes). Stir frequently to break up the anchovy paste and keep the garlic from burning. Add spaghetti to the pot of boiling water, and cook until al dente (a little chew left to the texture) or according to the instructions on the package. Add tomatoes (with juices), olives and capers to the large skillet, stirring to combine. Cook for 10 minutes (about the same amount of time needed to cook the spaghetti), stirring occasionally. Season sauce to taste with kosher salt.
Reserve 1/2 cup of the starchy pasta cooking water, then drain the spaghetti. Return the spaghetti to the pot along with the roasted cauliflower, tomato sauce, and a splash of starchy cooking liquid, tossing to combine. Add more starchy water as needed to bind the sauce with the spaghetti. Season to taste with additional kosher salt. Divide spaghetti between 4 dinner plates (or 2 with leftovers), and top with chopped parsley and black pepper. Serve with lemon wedges on the side.