Place 2 large boneless, skinless chicken breasts on cutting board. Use sharp chef's knife to trim chicken breasts, then slice each chicken breast in half horizontally, creating 4 thin chicken cutlets total. Set cutting board aside.
Add ½ cup all-purpose flour to shallow bowl. Place one chicken cutlet in flour and press lightly, then turn chicken cutlet over and press other side lightly into flour. Hold chicken cutlet over bowl and shake off excess flour, then place cutlet on large plate and set aside.
Repeat process with remaining chicken cutlets. Add more all-purpose flour as needed to thoroughly cover chicken.
When all cutlets have been dredged in flour, place large skillet on stovetop over medium-high heat. Add 2 tablespoons unsalted butter and 3 tablespoons extra-virgin olive oil to pan and continue heating, swirling or tilting pan as needed to cover entire surface.
When olive oil and melted butter begin to sizzle, lay 2 chicken cutlets in skillet, making sure cutlets do not overlap. Cook chicken 3 minutes or until browned on bottom, then flip cutlets over.
Cook cutlets 3 minutes more or until browned on other side. Once both sides are browned, transfer chicken cutlets to large plate and set aside.
Add remaining 2 tablespoons unsalted butter and 2 tablespoons extra-virgin olive oil to skillet. Repeat process with remaining chicken cutlets. Once cutlets have been cooked on both sides, transfer cutlets to plate and set aside.