Bring a large pot (at least 6 quarts) of water to a rolling boil and add 1 tablespoon kosher salt. The water should taste like seawater — this seasons the pasta from the inside out. Begin cooking the pasta 2–3 minutes shy of package al dente (it will finish in the sauce). Reserve 1 cup (240 ml) of the pasta cooking water before you drain.

While the water heats, prep the chicken: pat the breasts dry with paper towels (dry surface = better sear), season both sides generously with salt and black pepper, then slice each breast horizontally into 2 even cutlets if thick. This yields faster, more even cooking and more surface area for browning.

Heat a 12-inch skillet over medium-high heat and add 1 tablespoon olive oil. When the oil is shimmering, add the chicken in a single layer and sear without moving for 3–4 minutes until a deep golden crust forms (that Maillard browning gives savory flavor). Flip and cook 2–3 minutes more until just cooked through (internal temperature ~160°F/71°C). Transfer to a cutting board to rest 5 minutes, then slice thinly across the grain. Resting keeps the chicken juicy.

Reduce the heat to medium and wipe the skillet if there's a lot of burnt residue (leave any tasty browned bits). Add 2 tablespoons of the unsalted butter and melt. Add the minced onion and cook 3–4 minutes until translucent — this softens the onion and releases sweetness that balances the sauce. Add the garlic and cook 30–45 seconds — long enough to bloom the garlic but not brown it, which can make it bitter.

Make a light roux: push the onion-garlic to the side and add the remaining 1 tablespoon butter. Sprinkle the 2 tablespoons flour into the butter and cook, stirring constantly, for 60–90 seconds. Cooking the flour removes the raw taste and gives a nuttier base; don't let it brown significantly or your sauce will take on a toasty flavor.

Slowly whisk in the 1/2 cup chicken broth (or water) first to loosen the roux, then add the 1.5 cups whole milk in a steady stream while whisking to avoid lumps. Bring to a gentle simmer and cook 3–5 minutes until the sauce thickens and coats the back of a spoon. Simmering allows the starches to swell and thicken the sauce naturally.

Turn the heat to low and remove the skillet from direct heat. Add the grated Parmigiano-Reggiano and shredded cheddar a handful at a time, stirring to melt each addition before adding more. Cheese added off high heat melts smoothly and avoids grainy separation. Season with a pinch of freshly grated nutmeg, and adjust salt and pepper to taste.

Temper the egg yolk to add silk and body without scrambling: whisk the yolk in a small bowl, then slowly add 2–3 tablespoons of the hot sauce to the yolk while whisking constantly to gently raise its temperature. Then stir the tempered yolk into the sauce off the heat. This adds richness and shine without turning into scrambled egg.

Add the drained pasta and sliced chicken to the skillet. Toss gently to combine, using 1/4–1/2 cup of reserved pasta water a little at a time to loosen and marry the sauce to the pasta — the starchy water helps the sauce cling. Finish with the optional 1 tablespoon butter and chopped parsley for gloss and freshness.

Serve immediately, with extra grated Parmigiano-Reggiano on the side. Timing is key: serve while hot so the sauce is glossy and coats the pasta.