Make herb relish. Combine parsley, olive oil, chopped scallions, parmesan, lemon zest, lemon juice, and black pepper in a medium bowl. Stir until combined and set aside.
Prepare chicken. Butterfly chicken breasts and pound with a mallet until ¼-inch thick. Season both sides with lemon pepper and salt.
Place ¼ cup flour in a large shallow bowl. Coat chicken in flour, shaking off the excess.
Heat 2 tablespoons oil and 2 tablespoons butter in a large saute pan over medium-high heat. Lay the chicken in the pan and cook until browned, about 2 minutes. Flip chicken over and brown on the other side. Remove chicken and set aside.
In the same pan, heat another 1 tablespoon butter and 1 tablespoon oil. Add garlic and saute for 30 seconds. Add remaining 1 tablespoon flour, stirring to combine.
Add white wine and chicken broth and simmer to a boil. Reduce heat to low and simmer for 5 minutes, stirring occasionally. Add lemon juice and capers, stirring to combine. Season sauce with salt and pepper.
Add chicken back to pan, coating the chicken with the sauce. Simmer for 3-5 minutes or until chicken is hot. Remove from heat and transfer chicken to a serving platter. Spoon sauce on top and drizzle herb relish on chicken. Serve immediately.
Notes
If you can't find lemon pepper, substitute it with ½ teaspoon black pepper and 1 teaspoon lemon zest.