Bring a large pot of salted water to a rolling boil (approx. 1 tbsp salt per 4–6 liters). Salted water seasons pasta from the inside and helps the sauce cling.

While water heats, butterfly the chicken breasts (slice horizontally without cutting through, or halve into 2 thin cutlets). Pat dry and season both sides generously with salt and freshly ground pepper. Drying the surface encourages browning; even thin pieces cook quickly and stay juicy.

Heat a large skillet over medium-high heat. Add 1 tbsp olive oil and 1 tbsp butter. Once shimmering, add the chicken and sear without moving for 3–4 minutes per side until a deep golden crust forms and the internal temperature reaches 160°F (it will finish resting). Searing develops Maillard flavor — the brown crust gives savory depth to the whole dish. Transfer chicken to a plate to rest and tent loosely with foil.

Drop the pasta into the boiling water and cook until just shy of al dente (about 1–2 minutes less than package directions). Reserve 1 cup (240 ml) of pasta cooking water before draining. Pasta will finish cooking in the sauce; the starchy cooking water helps emulsify and thicken the sauce.

In the same skillet used for chicken, reduce heat to medium and add the remaining 2 tbsp butter. When melted, add the diced onion and cook until translucent, 3–4 minutes, to build sweet aromatics. Add the minced garlic and cook 30–45 seconds until fragrant — garlic cooks quickly and will turn bitter if overcooked.

Sprinkle the 2 tbsp flour over the softened onion and garlic, stirring to form a light roux. Cook 1 minute to remove the raw flour taste; this binds and thickens the milk into a smooth sauce rather than a grainy texture.

Gradually whisk in the 2 cups of whole milk, starting with a few tablespoons to loosen the roux, then pour the rest while whisking to prevent lumps. Bring to a gentle simmer and cook 3–5 minutes until the sauce thickens enough to coat the back of a spoon. Gentle simmering concentrates flavor and yields a velvety béchamel base.

Remove the skillet from the heat. In a small bowl, whisk the 2 egg yolks with 2 tablespoons of the hot sauce (temper the yolks) until smooth, then whisk the tempered yolks back into the skillet. Tempering prevents the yolks from scrambling and gives the sauce a silky, glossy finish.

Stir in the grated Parmesan off the heat until melted and incorporated. Taste and season with salt and plenty of freshly ground black pepper. Cheese adds umami and salt; adding it off heat prevents graininess and helps the sauce remain smooth.

Slice the rested chicken into 1/2-inch strips. Return the pasta (drained) to the large pot or a warm skillet and add the sauce. Toss to coat, adding reserved pasta water a little at a time (start with 1/4 cup) to adjust consistency — the starch in the water helps the sauce adhere without becoming heavy.

Add the chicken on top, toss gently to combine, and finish with lemon zest, chopped parsley, and an extra grate of Parmesan. Serve immediately so the sauce remains glossy and coats each strand.

If the sauce thickens while waiting, stir in a splash of warm milk or reserved pasta water to loosen it — gentle heat and liquid rescue a sauce that’s tightened up.