Bring a large pot of water to a rolling boil. Add 1 tablespoon salt. Cook fettuccine until al dente according to package directions. Reserve 1 cup of pasta water, then drain the pasta.
While pasta cooks, pat shrimp dry and season with salt and pepper. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add shrimp in a single layer, cook for 2 minutes per side until pink and opaque and the internal temperature reaches 145°F (63°C). Transfer to a plate.
Reduce heat to medium. In the same skillet, add remaining 1 tablespoon olive oil. Sauté minced garlic and red pepper flakes for 30 seconds until fragrant, stirring constantly. Pour in heavy cream, lemon zest, and lemon juice. Bring to a gentle simmer, then whisk in Parmesan until melted and smooth. Season with salt and pepper.
Add drained fettuccine and about 1/2 cup reserved pasta water to the skillet. Toss with tongs to coat the pasta in the creamy sauce, adding more pasta water as needed to reach a silky consistency. Stir in arugula and return the shrimp to the pan; toss gently until arugula is just wilted, about 1 minute.
Divide pasta among plates or a large platter. Top with extra lemon zest, a generous sprinkle of Parmesan, and a drizzle of olive oil. Serve immediately.