Bring a large pot of water to a rolling boil and season generously with salt (about 1 tbsp). Salted water seasons the pasta from the inside as it cooks — essential for well-seasoned pasta. Start the water first because it takes the longest.

Prep the chicken: pat 1 lb chicken breasts dry, season both sides with 1 tsp salt and 1/2 tsp freshly ground black pepper. Let rest at room temperature while you prepare other ingredients (this helps it cook evenly). If breasts are thick, butterfly or pound to an even 3/4-inch thickness for uniform searing and quick cooking.

Heat a large skillet over medium-high heat until shimmering, add 1 tbsp olive oil and 1 tbsp butter. Sear the chicken 3–4 minutes per side (depending on thickness) until golden and cooked through to 160–165°F internal temperature. Searing at high heat develops the Maillard crust which adds deep savory flavor. Transfer to a cutting board, tent with foil and rest 5 minutes, then slice thinly against the grain.

When the water returns to a boil, add the 12 oz fettuccine and cook to 1–2 minutes shy of package al dente (the pasta will finish cooking in the sauce). Reserve 1½ cups of the pasta cooking water before draining. Starchy pasta water is a chef’s secret for loosening and emulsifying sauces so they cling to the noodles.

While the pasta cooks, make a flavorful base: in the same skillet used for chicken, lower heat to medium and add remaining 1 tbsp olive oil and 3 tbsp butter. Add the finely diced small onion and cook gently until translucent (about 3–4 minutes) to sweeten and soften — this builds a savory-sweet backbone for the sauce. Add 3 minced garlic cloves in the last 30 seconds and cook until aromatic (don’t brown or garlic becomes bitter).

Sprinkle 3 tbsp all-purpose flour over the softened onion-garlic and stir to form a roux. Cook 60–90 seconds to remove raw flour flavor — this step thickens the sauce and helps it cling to the pasta. Toasting the flour briefly also creates a nuttier flavor.

Gradually whisk in 2 cups cold whole milk (pouring cold helps prevent lumps) a little at a time, scraping the pan to loosen the roux. Bring to a gentle simmer while whisking — simmering activates the thickening power of the flour and concentrates flavor. Simmer 4–6 minutes until the sauce coats the back of a spoon and has slightly thickened.

Reduce heat to low and stir in 1½ cups freshly grated Parmesan cheese in two additions, stirring until melted and fully incorporated. Add a pinch of nutmeg and 1/4 tsp red pepper flakes if using. Cheese adds savory umami and creaminess; add it off high heat to prevent grainy, separated sauce.

Temper the egg yolk: in a small bowl, whisk the single egg yolk. Add 2–3 tbsp of the hot sauce or reserved pasta water to the yolk, whisking constantly to slowly raise its temperature. Then whisk the warmed yolk back into the sauce off the heat. Tempering prevents the yolk from scrambling and the yolk gives the sauce a glossy, silkier texture and richer flavor.

Add drained pasta to the skillet with sauce (if your skillet is too small, return sauce to the large pot then add pasta). Toss with 1/2 cup of reserved pasta water, adding more as needed to reach a saucy, clingy consistency. The starch in the reserved water helps emulsify the sauce so it binds to the noodles instead of pooling.

Nestle the sliced chicken into the pasta and gently toss to combine. Taste and adjust seasoning with salt and plenty of freshly ground black pepper — finishing seasoning is important because cheese and pasta water can shift the salt balance.

Plate immediately, finish with a grind of black pepper, a sprinkle of the remaining Parmesan, chopped parsley for brightness, and an extra drizzle of olive oil or a small pat of butter for gloss. Serve hot so the sauce remains silky and not set.