Cook angel hair pasta according to package directions, drain and set aside.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat and season chicken with salt and pepper. Cook chicken pieces until golden brown and cooked through, about 6-8 minutes, then remove from skillet and set aside.
In the same skillet, add remaining olive oil and 2 tablespoons butter. Add bell peppers and cook for 3-4 minutes until slightly softened, then add minced garlic and red pepper flakes, cooking for 1 minute until fragrant.
Pour in white wine, chicken broth, and lemon juice. Bring to a simmer and cook for 3-4 minutes to reduce slightly.
Add remaining 4 tablespoons butter and stir until melted and sauce is smooth. Return chicken to the skillet and toss to coat.
Add cooked pasta to the skillet and toss everything together until well combined. Garnish with fresh parsley and serve immediately.