Slice the chicken breasts lengthwise into thin cutlets. No pounding is needed, but try to keep the pieces close in thickness so they cook evenly.
Season both sides of the chicken with salt, black pepper, and garlic powder.
Add the flour to a shallow plate. Lightly coat each piece of chicken in flour on both sides, then shake off the excess.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook for about 3–4 minutes per side, or until golden and cooked through. Transfer the chicken to a plate.
Lower the heat to medium and add the garlic to the same skillet. Cook for about 20–30 seconds, just until fragrant.
Pour in the chicken broth and add the capers. Scrape up the golden bits from the bottom of the skillet and simmer for about 3 minutes.
Turn off the heat, then stir in the lemon zest, fresh lemon juice, heavy cream, and butter.
Turn the heat back to low and cook gently for 2–3 minutes, stirring often, until the sauce becomes smooth and creamy. Do not let it boil hard.
Return the chicken to the skillet and spoon the creamy lemon caper sauce over the top. Warm for 1–2 minutes.
Finish with fresh parsley and serve warm.