Make Garlic Butter: Mix softened butter with minced garlic, parsley, and salt in a bowl. Set aside.
Cook Chicken: Slice chicken breasts horizontally to create thinner cutlets. Season with salt, pepper, and paprika. Heat olive oil in a skillet over medium heat. Cook chicken 5-6 minutes per side until golden and cooked through (165°F internal temperature). Rest 3 minutes, then slice into strips.
Butter the Bread: Spread garlic butter on one side of each bread slice, covering edges completely.
Assemble: On the non-buttered side of 4 slices, spread mayonnaise. Layer mozzarella, sliced chicken, cheddar, and Parmesan. Top with remaining bread slices, buttered side out.
Cook Sandwiches: Heat a skillet over medium heat. Cook sandwiches 3-4 minutes per side, pressing gently with a spatula, until golden brown and cheese is melted.
Serve: Let rest 1 minute, slice diagonally, and serve hot.