Bring a large pot of salted water to a boil and cook the fettuccine according to package directions, stopping 2 minutes before the package time so the pasta is slightly undercooked; reserve 2–3 tablespoons of pasta cooking water, then drain.
Heat a large skillet over medium heat and melt 1 tablespoon of the butter. Add the shrimp, season with salt and pepper, and cook 2–3 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, stirring constantly until the flour is cooked and no lumps remain, about 1 minute.
Slowly whisk in the heavy cream and whole milk, then remove the skillet from the heat and whisk in the egg yolk until smooth.
Return the skillet to low heat, add the grated Parmesan, and whisk until the cheese melts 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 cooked shrimp to the sauce, and toss gently until everything is evenly coated.
Serve immediately topped with chopped parsley and additional grated Parmesan if desired.