In a large nonstick skillet, melt 1 tablespoon of the butter in 1/2 tablespoon of the olive oil over high heat. Add 2 of the chicken breast halves, reduce the heat to moderate and cook until golden brown on the bottom, about 5 minutes. Turn the chicken and cook until browned on the second side and cooked through, 3 to 4 minutes longer. Transfer the chicken to plates and wipe out the skillet. Repeat with the remaining butter, oil and chicken. Season the tomato-avocado salad with salt and pepper, spoon it over the chicken and serve.