Season chicken breast pieces with salt and pepper, to taste. Dredge them in flour, shaking off excess.
In a skillet, heat the oil and pan-fry the chicken until golden brown, about 3 minutes per side. Do not crowd the skillet; work in batches adding oil as needed. When all the chicken is fried, drain most of the oil from the skillet, leaving only a thin coating in the pan.
Cook and stir minced garlic in the skillet for about 20 seconds (until fragrant). Pour in the chicken broth. Scrape and dissolve any brown bits at the bottom of the skillet. Add lemon slices, stir and bring to a boil. Stir occasionally, about 5 to 8 minutes, until the sauce reduces to about ⅔ cup.
Add capers and lemon juice. Simmer until sauce is reduced and slightly thickened, about 5 minutes. Add butter to the skillet and swirl it around until it is melted and fully incorporated. Add parsley. Remove skillet from heat and set aside.
Spoon sauce over chicken pieces and serve over rice or pasta.
