Pat chicken cutlets dry with paper towels, then season both sides evenly with salt and black pepper.
Place flour in a shallow bowl and dredge each cutlet, shaking off any excess coating thoroughly.
Heat olive oil and 2 tablespoons of butter in a large 12-inch skillet over medium-high heat until shimmering.
Add cutlets in a single layer and cook undisturbed for 3 to 4 minutes until deeply golden brown.⏱ 4 min
Flip cutlets over and cook the second side for another 2 to 3 minutes until cooked through.⏱ 3 min
Transfer cooked chicken cutlets to a warm plate and set aside while building the pan sauce.
Pour white wine into the skillet, scraping up any browned bits stuck to the pan with a wooden spoon.
Simmer wine for 2 minutes until reduced by half, then pour in the chicken broth, lemon juice, and drained capers.⏱ 2 min
Simmer sauce for 3 minutes over medium heat until slightly thickened and glossy.⏱ 3 min
Remove skillet from heat and swirl in the remaining 2 tablespoons of cold butter until fully melted and silky.
Stir in the chopped fresh parsley and return cutlets to the pan to coat with the sauce.
Spoon the warm lemon-caper sauce generously over the chicken and serve immediately with crusty bread or pasta.