Heat 2 tablespoons of olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Add the chicken pieces in a single layer; work in batches if needed. Sear for 3-4 minutes per side until golden brown and cooked through. Remove and set aside on a plate.

While the chicken cooks, bring a large pot of salted water to a boil. Add the angel hair pasta and cook until al dente, about 4-5 minutes. Reserve 1 1/2 cups of the pasta water before draining.

In the same skillet over medium heat, add the remaining 2 tablespoons of butter and 1 tablespoon of olive oil. Add the sliced red onion and bell peppers. Sauté for 3-4 minutes until softened.

Push the vegetables to one side. Add the minced garlic and red pepper flakes to the center. Cook for 30-60 seconds until fragrant, avoiding burning.

Pour in the white wine or broth. Scrape up the browned bits from the pan with a spoon. Reduce the liquid by about half.

Lower the heat to low. Stir in the reserved pasta water, lemon juice, lemon zest, and heavy cream if using. Stir until the sauce thickens slightly.

Return the chicken and any juices to the skillet. Add the drained pasta and toss to coat for 1 minute. Stir in half the parsley and half the Parmesan.

Garnish with remaining parsley and lemon zest if desired. Serve immediately with extra Parmesan.