Pat chicken breasts dry with paper towels. Set aside for 10 minutes to bring them to room temperature.
Place breasts between sheets of plastic wrap. Using the flat side of a meat tenderizer or a rolling pin, flatten breasts to an even thickness of 1 to 1.5 inches. Remove and discard top piece of plastic wrap.Use a pastry brush to apply 1 Tbsp of olive oil to the tops and bottoms of chicken pieces, and season each with some of the salt and pepper.
Heat a large stainless-steel skillet over medium-high heat. Add 1 Tbsp of olive oil to the pan and swirl to coat. When oil is shimmering, place chicken in pan, top (rounded) side down, and cook on medium-high without moving the pieces for 4 minutes. Use tongs to flip breasts over and cook for another 4 minutes. Turn heat down to low, cover skillet with its lid and let breasts cook for 5 minutes.
Check internal temperature of chicken with a thermometer. Chicken is done when the internal temperature is at least 163º. Remove chicken from pan and transfer to a plate or platter. Lightly cover plate with foil and let chicken rest while preparing gravy.
Make the Gravy
In a small bowl, make a slurry by stirring together cold chicken broth and cornstarch until smooth. Add butter to the skillet you cooked the chicken in and set over medium-high heat. When butter has melted, whisk in the cornstarch slurry. Bring mixture to a boil, then deglaze the pan by using a wooden spoon or spatula to scrape the bottom of the pan to loosen the browned bits.
Reduce heat to medium-low and simmer gravy for 3 minutes, or until thickened. Remove pan from heat. Spoon gravy over chicken breasts before serving.
Notes
Cooked chicken breasts can be kept in an airtight container in the fridge for up to 4 days, or in the freezer for up to 3 months.