Grate Parmesan, mince garlic, chop parsley, and measure water; melt 4 tablespoons unsalted butter in a small saucepan over low heat, skim and discard foam, and keep the clarified butter warm.

Heat 2 tablespoons olive oil in a 4- to 6-quart Dutch oven over medium heat, add minced garlic and cook 30–45 seconds until fragrant and translucent but not browned; add red pepper flakes if using.

Add 3 1/2 to 4 cups water and 1 1/2 teaspoons kosher salt; bring to a rapid simmer, then add the dry pasta and stir to submerge and separate the pieces.

Simmer uncovered, stirring every 2–3 minutes to prevent sticking; cook about 1–2 minutes less than package directions (start testing early for al dente).

When pasta is nearly al dente, reserve 1/2 cup of the starchy cooking liquid, then drain any obvious excess liquid and return the pasta to the pan over low heat.

Add 2 tablespoons cold unsalted butter and the warm clarified butter, tossing vigorously to emulsify; stir in 1/2 to 3/4 cup grated Parmesan until creamy, adding up to 2 tablespoons reserved pasta water if needed to loosen.

Stir in lemon juice (if using), chopped parsley, and season with black pepper and more salt to taste; serve immediately with extra Parmesan.

If adding shrimp, sauté 8 ounces peeled shrimp separately in 1 tablespoon oil 2–3 minutes per side until opaque and fold in at the end; for mushrooms, sauté 8 ounces sliced mushrooms after the garlic until browned before adding water and pasta.