Preheat oven to 375°F (190°C). Cook pasta in well-salted boiling water until 1–2 minutes shy of al dente. Drain, toss with a drizzle of oil, and set aside.
Pat shrimp dry and season with salt and pepper. Heat olive oil and half the butter in a large skillet over medium heat. Sear shrimp 1–2 minutes per side until just opaque. Remove and set aside — they will finish cooking in the oven.
Add remaining butter to the skillet. Cook garlic and red pepper flakes for 30 seconds until fragrant — do not let the garlic brown.
Pour in wine or broth, scraping up any browned bits. Add lemon juice and zest. Simmer for 2–3 minutes until slightly reduced.
Add drained pasta to the skillet and toss to coat. Stir in half the Parmesan and a generous handful of parsley. Fold the shrimp back through evenly.
Transfer to a greased 9×13 inch baking dish. Top with mozzarella then remaining Parmesan.
Bake for 15 minutes until cheese is melted, bubbling, and lightly golden. Rest for a few minutes, then garnish with parsley and a squeeze of fresh lemon before serving.