Position racks in the upper third and center of the oven and preheat the broiler. Line a rimmed baking sheet with parchment paper.
Place the Parmesan in a mini food processor and grind into a snow-like powder. You should have at least 1 1/4 cups. Set aside.
Set up a breading station with 3 pie plates or shallow bowls. Combine the flour, kosher salt, pepper and 1 tablespoon of the garlic powder in the first pie plate; stir with a fork.
Beat the eggs with 2 tablespoons water in the second pie plate.
Combine the panko with 1/2 cup of the ground Parmesan in the third pie plate and stir with a fork.
Press your palm down on a chicken breast and slice the chicken in half with a sharp knife parallel to the cutting board. Repeat with the remaining chicken breasts to make a total of 6 thin pieces. Cut each piece in half crosswise to make a total of 12 mini cutlets.
Working in an assembly line, dredge each cutlet in the seasoned flour and turn to coat both sides; tap off any excess. Dip the chicken cutlet into the beaten eggs and coat both sides, letting any excess drip off. Then dredge in the Parmesan breadcrumbs and turn to coat both sides. Transfer to the parchment-lined baking sheet.
Heat 1/2 inch of olive oil in a large skillet over medium heat until shiny. Working in batches if necessary, add the cutlets in a single layer and cook until golden brown and crispy, 2 to 3 minutes per side. Remove and discard the parchment from the baking sheet and replace it with a wire rack. Transfer the fried cutlets to the prepared baking sheet.
Make the compound butter: Combine the garlic with the butter, 1/4 cup of the ground Parmesan, and the remaining 1 tablespoon garlic powder in a small bowl. Stir together with a spoon until well combined.
Place the ciabatta rolls on a rimmed baking sheet. Spread the compound butter on the split sides. Broil on the center rack until golden brown and toasted, 1 to 2 minutes. Remove the buns to a work surface.
Remove the wire rack from the baking sheet and wipe the baking sheet clean. Place the cutlets on the baking sheet and top each with about 2 tablespoons of the tomato sauce and the mozzarella rounds. Broil on the upper rack until the tops are bubbly and the mozzarella is melted and golden, 2 to 3 minutes.
Meanwhile, heat the remaining sauce in a small saucepan over medium heat until warm. Divide among 4 ramekins.
To plate, carefully transfer the cutlets with a spatula to the bottom buns, sprinkle each with 2 teaspoons of the ground Parmesan and the basil. Close the sliders and serve with the warmed tomato sauce. Enjoy!
Feel free to use any mini or slider rolls you like.