Bring a large pot of salted water to a rolling boil.
Pat chicken dry and season with salt, pepper, Italian seasoning, and lemon pepper. Dredge in flour, shaking off excess.
Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Sear chicken in a single layer for 3-4 minutes per side until golden and cooked through (165°F). Transfer to a plate and tent with foil.
While chicken cooks, add angel hair pasta to the boiling water and cook according to package directions until al dente (3-4 minutes). Reserve ½ cup of pasta water before draining.
In the same skillet over medium heat, melt remaining 2 tablespoons butter. Add garlic and cook for 30-60 seconds until fragrant. Pour in white wine, scraping up browned bits, and let it reduce by half (3-4 minutes).
Add chicken broth and simmer for 2-3 minutes. Stir in lemon juice, lemon zest, and red pepper flakes. Add cherry tomatoes if using and cook 1-2 minutes.
Return chicken to the skillet. Add cooked angel hair pasta and toss to coat with the sauce. Add reserved pasta water as needed to reach desired consistency. Stir in half the Parmesan and parsley.
Serve immediately, garnished with additional Parmesan, black pepper, and red pepper flakes. Offer lemon wedges on the side.