In a large, deep skillet, heat 1 1/2 inches vegetable oil over medium high heat. The oil needs to be 360 degrees F for frying. If you do not have a frying thermometer, add a cube of white bread to hot oil. If the bread browns in a 40 count, the oil is ready.
Cover work surface with waxed paper. Arrange breasts on paper. Butterfly small chicken breasts, cut into flesh and across, but not through the breast, and open them up. The 6-ounce thin cut breasts may just be removed from package and arranged on waxed paper. Cover breast meat with a second sheet of waxed paper. Pound out cutlets to 1/4-inch thick with a small heavy skillet or a rubber mallet. Be careful not to tear meat. Roll up waxed paper and breasts and set aside.
Combine chopped herbs on cutting board with chopped garlic. Cut 4 (1 1/2 tablespoons) pieces of cold butter and coat each piece liberally with the herb garlic mixture.
In 3 disposable pie tins, set out flour, eggs beaten with a splash of cold water and bread crumbs. Uncover chicken and squeeze a wedge of lemon over breasts. Season chicken with salt and pepper. Place an herb-covered piece of butter on each piece of chicken. Wrap and roll cutlets tightly up and over the butter cubes. Discard pie tins used for breading. Secure stuffed chicken with toothpicks. Roll stuffed breasts in flour, then egg, then bread crumbs.