Put a baking sheet on the bottom rack of the oven and preheat to 500 degrees F (or as high as your oven will go). Position a wire rack over a second baking sheet, spray with nonstick spray, and set aside.
For the eggplant: Peel the eggplant and cut crosswise into 1/2-inch-thick medallions. Mix the panko and Parmesan together in a shallow dish and season with salt and pepper. In a separate shallow dish, beat the eggs. Dip only one side of the eggplant slices first in the beaten egg, then in the breadcrumb mixture to coat. Transfer to the wire rack.
Carefully remove the hot baking sheet from the oven and reduce the oven temperature to 350 degrees F. Pour the oil directly onto the hot baking sheet. Working quickly, place the eggplant slices, panko-side down onto the oiled baking sheet. Sprinkle the unbreaded side with salt. Put immediately back on the bottom rack of the oven and bake until the breaded bottoms are golden brown, 15 to 20 minutes. Remove eggplant from oven and flip over so the golden breaded side is facing up.
For the sandwich: Slice the rolls in half horizontally, keeping them all attached. Spread the bottom of the rolls with the tomato pesto and place on a baking tray.
Heat the butter in a small pan over medium-low heat. Add the garlic and a pinch of salt and cook, stirring, at a low simmer so that the garlic infuses the butter, about 5 minutes. If the garlic starts to brown too quickly, lower the heat. Set aside.
Layer half of the mozzarella on top of the pesto, then the eggplant, then the remaining mozzarella. Place the top of the rolls back on the sandwich. Brush the tops with the garlic butter. Bake the sliders in the oven until the cheese is melted and the bread is slightly toasted, 10 to 12 minutes.
For the sauce: In a saucepan, add the vodka sauce and heat over medium- to medium-low heat until warm. Whisk in the Calabrian chiles and cream. Set aside.
Remove the rolls from the oven and brush with more garlic butter. Cut into small sliders and serve with the vodka sauce alongside for dipping.