Using a rolling pin, pound the chicken-breast halves between 2 sheets of wax paper until they are the same thickness.
Heat the oil and garlic in a large skillet over medium-high heat until the oil shimmers and the garlic browns. (Don’t burn the garlic.)
Add the chicken and cook until golden, 3 to 5 minutes on each side.
Add 1 teaspoon salt, ½ teaspoon pepper, the vinegar, chicken broth, and rosemary.
Cover skillet, reduce heat to medium-low, and cook 3 minutes more or until the chicken is fork-tender. Transfer the chicken to a platter and keep warm.
Increase heat to high and boil sauce until it is thickened or syrupy, 3 to 5 minutes. Whisk in the butter and pour the sauce over the chicken.