Bring enough water to cover kumquats to a boil. Add whole kumquats and boil for 30 seconds. Strain and let cool. Slice kumquats in half lengthwise removing seeds as needed. Set aside.
In a Dutch oven or heavy-bottomed sauté pan, heat olive oil over medium high. Season chicken with salt and pepper. Add chicken to pan and brown on all sides, about 8-10 minutes total, basting with oil and juices as it cooks. Transfer to a plate. Pour off all but 1 tablespoon of fat and return pot to heat.
Add onion and cook until it begins to soften, approximately 1 minute. Add garlic and cook for an additional 30 seconds or until fragrant. Add wine and cook, scraping up brown bits, until syrupy, 1 to 3 minutes. Add bay leaves and broth. Return chicken, skin side up, to the pan, and add kumquats and olives.
Bring to a boil; reduce to a simmer, cover and cook 20 minutes. Uncover and cook an additional 10 minutes, until chicken is cooked through.