Butterfly the chicken by slicing the chicken breasts in half lengthwise (like a book). Place the chicken in between plastic wrap and use a meat mallet to pound the chicken until it is about ¼-inch thick. This is important, as skipping this step will result in undercooked, tough chicken. You should have 6 cutlets when finished butterflying. Season the chicken with black pepper and salt.
Heat a large skillet over medium heat and add 2 tablespoons of the olive oil.
Sear the chicken on one side for 5-6 minutes. Flip and sear the other side for another 5-6 minutes or until both sides are browned. Remove from the pan and set aside on a plate.
Add the remaining olive oil to the pan. Add the onions, garlic, and rosemary to the pan. Sauté for 3 minutes or until translucent and slightly browned. Scrape the brown bits off the bottom of the pan.
Stir in the peppers and cook for 1 minute.
Add the chicken broth and lemon juice to the pan. Bring to a simmer and add the chicken back to the pan. Flip the chicken so that both sides are covered in the sauce. Put the lid on the pan and adjust the temperature to medium-low. Simmer for 20 minutes.
Once the chicken has cooked for 20 minutes, remove the chicken from the pan and place in a serving dish. Heat the sauce on high for 2-3 minutes, stirring constantly—this helps the sauce thicken.
Pour the sauce over the top of the chicken before serving.