Preheat the oven to 400 degrees F. Line a baking sheet with parchment.
Split the slider rolls in half. Place the bottoms cut-side up on the prepared baking sheet and the tops cut-side down. Melt the butter in a small skillet over medium-low heat. Add the garlic and cook until softened but not colored, 1 to 2 minutes. Brush the garlic butter on the cut sides of the bottom rolls and the tops of the top rolls. Sprinkle all with 1/4 cup Parmesan. Season with salt and pepper. Bake until the edges are golden and the cheese is crisp, about 5 minutes.
Slice the chicken breasts in half lengthwise to make 6 thinner cutlets, then cut each cutlet in half crosswise to make 12 bun-size pieces of chicken. Season with salt and pepper. Spread the flour in a shallow bowl, beat the eggs in a second shallow bowl and spread the breadcrumbs in a third. Dredge each chicken piece first in the flour, then in the eggs and then in the breadcrumbs.
Heat about 1/2 inch of vegetable oil in a large nonstick skillet over medium heat. Test the oil by dropping in a few breadcrumbs; when the oil sizzles, it's ready. Cook the chicken in batches, until the coating is crisp and browned and the chicken is just cooked through, about 2 minutes per side. Drain on a wire rack in a baking sheet.
Remove the bun tops from the baking sheet. Spoon half of the marinara in the indent of each bun bottom and top with a piece of chicken. Top with the remaining marinara and the sliced mozzarella. Sprinkle with the remaining 1/4 cup Parmesan. Bake until the cheese is melted and bubbly, about 5 minutes. Add a basil leaf or two to each slider before topping with the buns. Serve immediately.