Bring a large pot of water to a boil, season it with the kosher salt, and add the spaghetti. Cook according to the package directions until just al dente, then reserve 1/2 cup of the pasta cooking water before draining.

While the pasta cooks, pat the shrimp dry with paper towels and season them evenly with the fine sea salt and black pepper.

In a large skillet over medium-high heat, heat 1 tablespoon of the olive oil and 1 tablespoon of the butter until the butter is melted and just starting to foam.

Add half of the seasoned shrimp to the hot skillet in a single layer and cook for 1 to 2 minutes per side, just until the shrimp are opaque and lightly golden in spots. Transfer the cooked shrimp to a plate and repeat with the remaining shrimp, adding them to the same plate once cooked.

Reduce the heat to medium, add the remaining 1 tablespoon olive oil and 1 tablespoon butter to the same skillet, and let the butter melt.

Add the minced garlic and red pepper flakes to the skillet and cook, stirring constantly, for about 30 to 60 seconds until the garlic is fragrant but not browned.

Pour in the reserved 1/2 cup of pasta cooking water, stir to combine, and let the mixture simmer for 1 to 2 minutes to slightly reduce and form a light sauce.

Add the drained spaghetti directly into the skillet and toss well in the garlicky sauce so the pasta is evenly coated.

Return the cooked shrimp and any accumulated juices on the plate to the skillet, then add the lemon juice and lemon zest, and toss gently to combine everything.

Sprinkle in the grated Parmesan cheese and chopped parsley, toss again until the cheese melts lightly into the sauce, then taste and adjust seasoning if needed before serving.