Fill a large pot with water, add a generous pinch of salt, and bring to a boil. Add the linguine and cook according to package instructions until al dente (usually about 9-11 minutes). Reserve 1 cup of pasta water, then drain the rest and set the linguine aside.
While the pasta cooks, heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Add the shrimp in a single layer and season with salt and pepper. Cook shrimp for about 2 minutes on each side, until pink and opaque. Remove shrimp from the skillet and set aside.
In the same skillet, reduce heat to medium and add the minced garlic. Sauté for about 30 seconds until fragrant but not browned. Pour in the heavy cream and bring to a gentle simmer. Stir in the basil pesto and half of the Parmesan cheese until the sauce is smooth and creamy.
Add the cooked linguine and shrimp back into the skillet with the sauce. Toss everything together gently, adding reserved pasta water a splash at a time to loosen the sauce as needed. Taste and adjust salt and pepper.
Plate the pasta and garnish generously with the remaining Parmesan and fresh basil leaves. Serve immediately for the best flavor and texture.