Bring a large pot of salted water to a boil and cook the fettuccine according to package instructions, then drain 2 minutes earlier than recommended (reserve 2–3 tablespoons of pasta cooking water) and set aside.
Heat a large skillet over medium heat and melt 1 tablespoon of the butter.
Add the shrimp to the skillet, season with salt and pepper, and cook 2–3 minutes per side until pink and opaque; transfer shrimp to a plate.
Add the remaining 2 tablespoons butter to the skillet and sauté the minced garlic until fragrant and lightly golden, about 30–60 seconds.
Sprinkle the flour over the garlic and butter, stirring constantly until the flour is cooked and no lumps remain, about 1 minute.
Slowly whisk in the heavy cream and whole milk until smooth and heated through.
Temper the egg yolk by whisking a small amount of the hot cream mixture into the yolk, then whisk the yolk mixture back into the sauce; continue cooking gently until slightly thickened.
Stir in the grated Parmesan until melted and the sauce is smooth; if the sauce is too thick, loosen with 2–3 tablespoons reserved pasta water.
Lower the heat, add the cooked fettuccine and shrimp to the sauce, and toss with tongs until everything is evenly coated and heated through.
Serve immediately garnished with chopped parsley and additional Parmesan if desired.