Place each chicken breast between two sheets of plastic wrap and gently pound to an even 1/2-inch thickness. Season both sides with salt and pepper.
Spread flour on a shallow plate. Dredge each chicken piece in flour, shaking off the excess. This light coating adds a subtle crust and helps thicken the sauce.
In a large skillet, heat olive oil over medium-high heat. When the oil shimmers, add the chicken and cook about 2-3 minutes per side, until golden and cooked through. Transfer to a plate and tent with foil to keep warm.
Reduce heat to medium. In the same skillet, melt butter. Add the wine (or chicken broth) to deglaze, scraping up any brown bits from the bottom for flavor. Simmer for 2-3 minutes until reduced slightly.
Stir in the chicken broth, lemon juice, and capers. Bring to a gentle simmer and cook for about 2 minutes, allowing the sauce to reduce and become glossy.
Return the chicken to the pan and spoon the sauce over the pieces. Cook for 1-2 minutes more to rewarm the chicken and meld flavors.
Taste and adjust the acidity with a touch more lemon juice or a pinch of salt if needed. Sprinkle with chopped parsley and a light grind of pepper. If you like, a small grating of parmesan on top adds a salty finish.
Serve immediately. This dish shines with a side of spaghetti, sautéed greens, or roasted potatoes to soak up the lemony sauce.