Bring a large pot of heavily salted water to a boil over high heat. Meanwhile, finely grate the zest of 1 medium lemon (about 2 teaspoons), then juice the lemon (about 3 tablespoons). Finely grate 1 1/2 ounces Parmesan cheese (about 2/3 cup), or measure scant 1/2 cup store-bought grated. Finely grate or mince 2 garlic cloves. Pat 1 1/2 pounds raw peeled and deveined jumbo shrimp dry with paper towels and season all over with 3/4 teaspoon of the kosher salt.
Add 1 pound dried fettuccine pasta to the boiling water. Cook, stirring every few minutes and separating any noodles that are sticking together, for 1 less minute than al dente according to the package directions, 9 to 10 minutes. Meanwhile, make the sauce.
Heat 1 tablespoon olive oil in a large frying pan over medium-high heat until shimmering. Add half of the shrimp in a single layer and cook until starting to brown on the bottom and they release easily from the pan, about 2 minutes. Flip the shrimp and cook until just cooked through, about 1 minute more. Transfer to a bowl or plate.
If the pan is dry, add a little bit more olive oil. Add the remaining shrimp and repeat cooking and transferring to the bowl or plate.
Reduce the heat to medium. Add the garlic and 2 tablespoons unsalted butter to the pan and cook until the butter is melted, about 45 seconds. Add 1 cup heavy cream and bring to a simmer, scraping up the browned bits from the bottom of the pan. Add the Parmesan and remaining 1 teaspoon kosher salt, and stir until the cheese is melted. Remove the pan from the heat.
When the pasta is ready, reserve 1 cup of the pasta water. Drain the pasta, then add the pasta to the sauce. Add the shrimp and any accumulated juices on the plate, and 1/2 cup of the pasta water. Cook over medium heat, tossing constantly, until the sauce thickens slightly and coats the pasta, about 2 minutes. Toss in more pasta water a tablespoon at a time as needed to reach the desired sauce consistency.
Turn off the heat. Add the lemon zest and 1 tablespoon of the lemon juice, and toss to combine. Taste and season with more kosher salt and lemon juice as needed. Garnish with chopped fresh parsley leaves if desired.