Heat a small amount of oil in a pan over medium heat. Place onions in pan and cook until the onions are soft and translucent. Turn the heat down to low and leave, cooking, until the rest of the recipe is done, stirring occasionally.
Place the chicken breast halves, one at a time, between layers of wax paper or plastic wrap. Pound each breast half so that it is an even 1/2-inch thick. Sprinkle the chicken with salt and pepper, to taste. Set aside.
Heat the 2 teaspoons of olive oil in an extra-deep, 12-inch nonstick skillet over medium heat. Add the chicken to the skillet and cook for 4-5 minutes on the first side until golden brown. Meanwhile, cut the lime in half and cut 1 of the halves into 4 wedges. Set the wedges aside.
Turn the chicken over and squeeze the juice from the lime half evenly over the chicken. Continue to cook for 4 to 6 minutes, or until the chicken is no longer pink in the center. Put the chicken on 4 serving plates.
Add the garlic to the hot skillet and stir constantly until the garlic is lightly browned, about 1 minute. Remove the skillet from the heat, add the carmelized onions to the browned garlic, and mix together. Top each piece of chicken with 1/4 cup onion and garlic mixture. Serve at once, garnished with reserved lime wedges.