Place chicken breasts between two sheets of plastic wrap. Flatten with a meat mallet or rolling pin until the breasts are ¼ in. thick.
Sprinkle ½ tsp. salt and ½ tsp. black pepper (total) over each side of the chicken breasts. Set aside.
Combine crushed crackers, 2 c. flour, baking powder, 1 ½ tsp. salt, 1 tsp. black pepper, and crushed red pepper.
Whisk together 1 ½ c. milk and eggs. Dredge each chicken breast in the cracker-flour mixture, dip in the milk mixture, and dredge in the cracker-flour mixture once more.
Preheat oven to 200F. Pour ½ in. of oil into a skillet. Heat to 360F. Fry chicken for 3-5 minutes on each side or until golden brown. The chicken should reach an internal temperature of 165F. Once done, place the chicken on a wire rack on a baking sheet. Keep warm in the low-temperature oven.
Carefully drain hot oil, reserving the cooked bits and 2 tbsp. of drippings in the skillet.
Whisk together the remaining ½ c. flour, 2 ½ tsp. salt, 1 tsp. black pepper. Add to the chicken drippings and grease and cook for 1-2 minutes. Add milk and cook over medium-high heat, whisking constantly, until the gravy has thickened. If you want a thinner gravy, add more milk.
Serve fried chicken breasts with gravy on top and sprinkle with a small amount of additional black pepper, if desired.