Using a sharp knife, cut the chicken breasts in half lengthwise. Thinner breasts will be much easier to pound down to 1/4" thickness. Small chicken breasts are thinner and don't need to be cut in half. Place the breasts between two sheets of plastic wrap and gently pound them until they are about 1/4" thick. Set aside.
To prepare the egg wash, in a small bowl, whisk together milk (or water) and eggs. Set aside.
In a separate bowl, combine the broth, wine, lemon juice, one teaspoon each cornstarch and salt, and whisk until all ingredients are blended together. Set aside.
Place the rest of the cornstarch in a shallow dish for dredging, add two teaspoons salt and one teaspoon black pepper, whisk and set aside.
Pat the cutlets dry with paper towels and season with salt and pepper. Dredge each cutlet in cornstarch, shaking off excess, then dip in egg mixture to coat, letting excess to drip off, then dredge in cornstarch again and set aside on a large platter or a baking sheet in a single layer.
Heat two tablespoons of butter in a large skillet over medium-high heat. Carefully add 4 cutlets and cook until golden brown and cooked through, about 2 to 3 minutes per side. Transfer to a platter. Repeat this step with the remaining four cutlets.
Add the garlic and the capers to the skillet and cook for 30 seconds. Add the broth mixture and bring to a boil and cook for about 5 minutes, until reduced by half.
Meanwhile, if you want the sauce to be thicker, knead 2 tablespoons of butter with 1 teaspoon of flour. Otherwise just use plain butter. Add the butter to the pan and stir well until the butter is incorporated into the sauce. Remove from heat and set aside.
Place the chicken cutlets on a serving platter and spoon the sauce over the chicken. Garnish with chopped parsley and serve.