Season chicken with salt and pepper, then dredge the pieces in flour. Shake off excess flour. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add chicken, working in batches, and sauté until chicken is cooked through about 8 to 10 minutes. Wipe the skillet clean between batches. Transfer chicken to a plate.

Reduce heat to low and add garlic and butter. Cook until fragrant. Add white wine, bring to a simmer and reduce slightly about 1 minute. Add chicken stock, lemon juice, and capers. Bring to simmer and scrape the pan with the back of a wooden spoon to loosen the cooked chicken bits. Stir in heavy cream, if using. Return chicken to the pan and add cooked angel hair. Add grated Parmesan and toss until pasta is coated in sauce.

Garnish with parsley and serve immediately.