Pat chicken dry with paper towels and season with salt and pepper. heat oil in large skillet over medium-high heat until just smoking. Cook chicken until golden brown, about 5 minutes per side. Transfer to plate.
Add onion to fat in skillet and cook until lightly browned, about 3 minutes. Stir in garlic and cook until fragrant, about 1 minute. Add peppers, sugar, broth, and browned chicken along with any accumulated juices and bring to a boil. Reduce heat to medum and simmer until chicken is cooked through, about 5 minutes.
Transfer chicken to serving platter, and tent with foil. Return skillet to high heat and simmer until sauce is slightly thickened, about 5 minutes. Off heat, whisk basil and butter. Season with salt and pepper. Pour sauce over chicken. Serve.