Bring salted water to a boil for angel hair pasta. Heat 1 tbsp olive oil in large sauté pan over medium heat, add prosciutto and sauté until lightly brown. Add remaining olive oil, 1 tbsp butter, garlic, capers, artichoke hearts and cook until garlic is fragrant, about 3 minutes.
Scrape up any browned bits on pan bottom and add chardonnay to deglaze. Add lemon rounds and simmer over medium heat until liquid is reduced by half. Add remaining butter, stirring over low heat until incorporated. Remove lemon rinds.
Place cooked chicken in the sauce to warm through, adding salt and pepper to taste. Meanwhile cook angel hair pasta according to package instructions and drain. Pour chicken and sauce over pasta and sprinkle with parmesan cheese and parsley.
Serve immediately with additional parmesan cheese on the side.