Prep the chicken. Use a thin, sharp knife to slice each chicken breast in half lengthwise to create two thin fillets. Place the chicken breasts between two pieces of plastic wrap on a cutting board, and gently pound to about ⅜-inch with a meat tenderizer.
Bread the chicken. Add ⅓ cup flour to a shallow bowl. In another bowl, whisk 2 eggs with ¼ cup milk, and season with ½ teaspoon salt and ½ teaspoon black pepper. Add 1 cup dry bread crumbs to another shallow bowl. One at a time, lightly dredge the chicken breasts first in the flour, then the egg mixture, and then in the bread crumbs, shaking off excess, and place on a wire rack. Repeat with the remaining chicken breasts. Let the coated chicken rest at room temperature for 20 minutes to help set the coating.
Fry the chicken. Heat ¼ cup canola oil and 1 tablespoon butter in a heavy skillet over medium-high. Fry the chicken in batches, spooning the hot oil over the top of the chicken from time to time, until golden brown and crispy, 2-3 minutes on each side, flipping so the chicken breading doesn’t burn. Cook until the internal temperature reaches 160°F then transfer to a platter and sprinkle with flaky salt. Wipe out any crispy or burned bits or discard the oil altogether then add the remaining ¼ cup of oil and 1 tablespoon butter to the pan and repeat with the remaining chicken breasts. Serve with a squeeze of fresh lemon juice.