Rinse the chicken and pat dry with paper towels. Sprinkle with salt and pepper. Heat the olive oil over medium heat in a large saucepan. When the oil is hot, add the chicken and cook, turning occasionally, until the skin is evenly browned, about 15 minutes. Remove the chicken and set on a plate.

Sauté the onion and garlic in the saucepan over medium heat until lightly golden and fragrant. Pour in the wine and add the bay leaves. Cook over high heat, stirring occasionally, until the liquid has reduced to a syrupy consistency. Transfer the chicken, skin side up, back into the saucepan. Pour enough chicken broth into the pot to cover the chicken 2/3 of the way up. Fold the kumquats and green olives into the broth. Cover and simmer over medium-low heat for about 40 minutes, until the sauce has thickened and the chicken is tender and cooked through. Season to taste with salt and pepper, and serve.