Trim fat and membranes from the chicken breasts and cut each breast in half to make 6 pieces; place one piece at a time inside a zip-top bag and pound to an even 1/2-inch thickness.
Season both sides of each chicken piece lightly with salt, pepper, and poultry seasoning.
Heat 2 tablespoons olive oil in a 12-inch nonstick skillet over medium-high heat until hot.
Add the rinsed, dried capers and fry until they puff and crisp slightly, about 60–90 seconds; remove with a spoon to a paper towel, leaving the oil in the pan.
Return the pan to medium-high heat, add chicken pieces and cook with the top side down until nicely browned, about 3–4 minutes; flip and cook the second side until just cooked through, about 1–2 minutes.
Transfer the cooked chicken to a serving dish, browned side up, and keep warm if desired.
Add the minced garlic to the hot pan and cook 15–30 seconds until fragrant, then add the chicken stock and scrape the pan to dissolve browned bits.
Cook the stock until reduced by about half, approximately 2 minutes.
Stir in the lemon juice and optional sweetener, then remove the pan from the heat and stir in the butter pieces until melted and the sauce is glossy.
Stir half the fried capers and the chopped parsley into the sauce, pour the sauce over the chicken, and garnish with the remaining capers.