Position rack in the upper third of the oven and preheat to 350 degrees F. Place garlic cloves in a small, ovenproof pan and drizzle with 2 tablespoons of olive oil. Toss to coat evenly.
Place pan in oven and roast until garlic is light golden brown, about 15 minutes. Remove promptly.
Meanwhile, bring a large pot of water to a boil. Add 2 tablespoons salt and cook pasta until al dente. When the pasta is ready, reserve 1 and 1/2 cups of the pasta cooking liquid for use later in the recipe. Drain the remaining water, then set pasta aside, tossing it with a bit of olive oil to prevent noodles from sticking.
When the garlic has finished roasting, transfer the cloves and any oil left in the pan to a food processor or blender. Add 1/4 cup of Parmesan, then pulse a few times to combine. With the food processor or blender running, slowly pour in the remaining 1/3 cup olive oil and blend until mostly smooth.
Heat a skillet large enough to hold the pasta over medium. Add the garlic –olive oil mixture and red pepper flakes. Cook for 30 seconds, then add the reserved pasta cooking liquid. Bring to a boil, then lower the heat and add the remaining 2 teaspoons salt. Simmer for 5 minutes, until reduced by about 1/3.
Add reserved pasta to the skillet and toss to coat. Remove pan from heat, add remaining 1/2 cup parmesan cheese and parsley, then toss well. Allow the mixture to rest for 5 minutes so that the noodles absorb the sauce. Serve warm topped with additional Parmesan.
Notes
TO STORE: Leftovers may be stored in a container with a tight-fitting lid for up to 5 days.
TO REHEAT: To reheat, warm in a nonstick skillet, covered, over medium-low heat with a splash of water. Once the pasta starts steaming, uncover, and toss until heated through. Alternatively, you may also warm in the microwave with a splash of water to rehydrate the pasta, covered, until steaming.