Position a rack in the center of the oven and heat the oven to 425°F.
Make a lengthwise horizontal slice almost through each chicken breast and open each up like a book.
Flatten the chicken with a meat mallet until it is 1/4" thick.
put the cheese, breadcrumbs, 3 Tbsp capers, lemon zest, and 1 Tbsp parsley in a mini chopper of food processor and pulse a few times to combine.
Sprinkle the mixture on top of the chicken breasts.
Fold each breast closed and secure with toothpicks.
Sprinkle the breasts with salt and pepper.
Heat 1 Tbsp butter and the oil in a large (12"), heavy-duty, oven-proof skillet over medium-high heat until the butter melts and starts to foam, about 2 minutes.
Add the chicken and cook, without touching, until it browns and easily releases from the pan, about 2 minutes.
Turn the chicken and cook the other side until browned, about 2 more minutes.
Add the garlic and the remaining 1 Tbsp capers to the skillet, transfer the pan to the oven, and roast uncovered until the chicken cooks through (165°F), about 8 minutes.
Transfer the chicken to a serving platter and tent with foil.
Set the skillet over medium-high heat; add the chicken broth and cook, scraping the bottom of the pan with a wooden spoon to loosen any brown bits, until it reduces by half, about 2 minutes.
Remove from heat and whisk in 2 Tbsp of the lemon juice and the remaining 2 Tbsp of butter.
Taste and add more lemon juice, salt, and pepper, if needed.
Serve the chicken drizzled with the butter sauce and sprinkled with the remaining 1 Tbsp parsley.