Pound chicken to an even thickness using a meat mallet or rolling pin. OR if the pieces are huge, cut into cutlets. Season both sides with salt and pepper.
Beat the egg in a shallow dish (I like to use a pie plate). Take a second shallow dish and add the flour. Take a third shallow dish and for breadcrumbs.
To assemble: take the chicken breasts, one at a time, and dredge into the flour mixture, coating both sides, then dip into the egg mixture. Lastly, coat with breadcrumbs. Transfer coated chicken pieces to a sheet pan.
Preheat a large skillet (12-13 inches) over medium heat. Add the olive oil. Add the chicken without overcrowding the pan. You may have to do this in batches depending on the size of your cutlets. Cook for 5 minutes per side, or until each side is a deep golden brown and the chicken is cooked through. Transfer to a cooling rack. Set aside.
Preheat oven to 350 degrees. Slice the sliders in half and place the bottom half on a baking sheet cut side up. Bake until lightly toasted, about 7 minutes.
Completely cover one half with mozzarella and grated Parmesan on the other half. Bake until cheese is melted, about 5 minutes.
Spread vodka sauce on top of mozzarella. Add chicken to the bottom half of the buns. Drizzle with pesto. Add the bun tops.
Cut sliders into individual sliders and serve.