Bring a large pot of salted water to a boil and cook the fettuccine according to package directions, reducing the cook time by 2 minutes so the pasta is slightly under al dente; reserve 2–3 tablespoons of pasta cooking water, then drain and set aside.
Meanwhile, 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 to 3 minutes per side until pink and opaque; transfer the 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 and cook, stirring, for 1 minute to remove the raw flour taste and form a roux.
Slowly whisk in the heavy cream and milk until smooth, then remove the skillet from the heat briefly and whisk in the egg yolk until combined.
Return the skillet to low heat, add the grated Parmesan, and whisk until the cheese melts and the sauce is smooth; if the sauce becomes too thick, stir in 2–3 tablespoons reserved pasta water to loosen it.
Lower the heat, add the cooked fettuccine and shrimp to the sauce, and toss gently until everything is evenly coated and heated through.
Serve immediately garnished with chopped parsley and extra Parmesan if desired.
