Heat 1 tbsp olive oil and 1 tbsp butter in a large skillet over medium-high heat. Season the sliced chicken with 1/2 tsp salt and 1/4 tsp black pepper, then sear for 3 to 4 minutes per side until golden and cooked through to 165°F (74°C). Remove and set aside.

Boil the 12 oz angel hair pasta in heavily salted water for 3 to 4 minutes until just al dente. Reserve 1/2 cup pasta water, then drain.

In the same skillet over medium heat, add the remaining 1 tbsp olive oil. Cook the 6 minced garlic cloves and 1/2 tsp red pepper flakes for 60 seconds, then add the 1/2 cup white wine and reduce for 2 minutes. Pour in the 1/2 cup chicken broth and 3 tbsp lemon juice, simmer 3 minutes, then stir in the remaining 3 tbsp butter until melted.

Add the drained pasta to the skillet and toss with tongs for 1 to 2 minutes over medium heat, adding pasta water as needed. Stir in the 1/2 cup Parmesan and most of the parsley until the sauce is glossy and coats the pasta.

Plate the pasta, lay the seared chicken on top, and garnish with the remaining parsley and lemon rounds.