Cut each chicken breast in half lengthwise so you have 2 thin halves of chicken. Place one piece in a large plastic bag or under plastic wrap and pound with a tenderizer or heavy pot until chicken is about ¼" thick. Repeat with remaining pieces until you have 4 cutlets. Pat dry; season both sides with salt.

Place flour in a large, shallow bowl. In another shallow bowl, beat eggs to blend. In a third shallow bowl, combine bread crumbs, oregano, and ½ cup Parmesan.

In a large skillet over medium heat, heat oil until a deep-fry or instant-read thermometer registers 350°. Line a baking sheet with parchment and set a wire rack inside. Carefully lower 2 cutlets into hot oil. Fry until undersides are golden and crispy, about 2 minutes. Carefully turn cutlets and continue to fry until other side is golden and crispy, about 2 minutes more. Transfer cutlets to prepared rack. Repeat with remaining cutlets.

In a small saucepan over medium heat, cook hot sauce and butter, stirring occasionally, until butter is melted. Spoon sauce over tops of chicken, spreading all the way to the edges; reserve remaining sauce for serving.

Place a rack in top third of oven; heat broiler to high. Sprinkle mozzarella and remaining ½ cup Parmesan over chicken.

Broil, watching closely, until cheese is melted, bubbling, and lightly golden, 2 to 4 minutes, depending on strength of your broiler.

Transfer chicken to a platter. Top with chives. Drizzle with ranch dressing and reserved sauce.