Prep and season the chicken: Cut the chicken breasts into 1-inch pieces. Pat them dry with paper towels — this is crucial for browning. Season generously with salt and pepper on all sides. Don't skimp on the salt here — it seasons the whole dish.
Sear the chicken: Heat 2 tablespoons of butter in your skillet over medium-high heat. When it's hot and shimmering (not smoking), add the chicken in a single layer. Cook for 3-4 minutes without moving it — let that golden crust form. Flip and cook 2-3 more minutes until cooked through. Transfer to a plate and set aside. (Photo tip: You're looking for deep golden-brown patches on the chicken — not pale white. If it's browning unevenly, your pan wasn't hot enough when the chicken went in.)
Drop the pasta: While the chicken cooks, add the angel hair to the boiling water. Cook for 1 minute LESS than the package directions — usually about 3 minutes total. Before you drain, scoop out 1/2 cup of pasta water and set it aside. Then drain the pasta and set it aside. (Photo tip: The pasta should still have a thin white core when you bite it — it'll finish cooking in the sauce.)
Bloom the garlic: Reduce the skillet heat to medium-low. Add the remaining 2 tablespoons of butter. Once it melts, add the minced garlic and a pinch of red pepper flakes. Cook for 30-45 seconds, stirring constantly, until the garlic is fragrant but not brown. Burnt garlic = bitter sauce. This is where the whole kitchen starts smelling like the restaurant — my kids come running.
Deglaze with wine: Pour in the white wine. Increase the heat to medium-high and bring it to a simmer. Scrape the bottom of the pan with a wooden spoon to loosen all those browned bits — that's gold. Let it simmer for 3-4 minutes until it reduces by about half. It should smell like wine, not vinegar.
Build the sauce: Add the chicken broth, heavy cream, and lemon juice. Stir to combine. Let it simmer for 2-3 minutes until it thickens slightly — you want it to coat the back of a spoon. Taste it here and add salt and pepper if needed. The sauce should taste savory with a bright finish from the lemon.
Finish everything together: Add the drained pasta and the cooked chicken back to the skillet. Toss to coat everything in the sauce. Sprinkle in the Parmesan and toss again. If the sauce seems too thick, add a splash of that reserved pasta water. If it's too thin, let it simmer another minute. What you're looking for: every strand of pasta is glossy with sauce, not sitting in a puddle.
Serve immediately: Divide among bowls, garnish with fresh parsley and a little more Parmesan. Eat it while it's hot — angel hair doesn't wait for anyone.