Using three different pans, separate out the flour into one, the eggs into another and the panko bread crumbs, Italian seasoning, salt and pepper into the the last. Use a fork to scramble the eggs and set aside.
Using a meat mallet (or the back of a pan covered in plastic wrap) smash the chicken breasts until they reach about ½" thickness. Pour the oil into a heavy bottomed pan and put the heat over medium high.
While the pan heats, coat one of the flattened chicken breasts in the flour, then transfer it to the eggs and coat both sides with the eggs. Lastly, transfer the chicken to the panko and make sure the chicken is throughly caked.
Once the oil has gotten hot enough that a drop of water sizzles when added, carefully add in the chicken. Cook for 5 minutes or until the breading is golden brown and flip and repeat on the other side for another 5 minutes or until chicken is cooked through all the way. Repeat steps 3 and 4 with all three chicken breasts, adding oil to the pan as necessary to continue frying. Once the chicken is cooked, transfer it to a paper towel lined plate and set aside.
Preheat oven to 400°F.
Lightly oil the bottom of a 9x13 pan. Use a serrated knife to cut the sliders in half horizontally and place the bottom layer in the pan. (Note: You can separate the sliders like I did in the photos if you want, but it isn't necessary.)
Spoon a large spoonful of the pesto onto each of the bottom buns and spread around as needed. Layer the chicken breasts on top, followed by the sauce, sliced fresh mozzarella, and the top buns.
Brush the tops of the sliders with the melted butter and sprinkle the parmesan on top. Cover the pan with aluminum foil and bake for 10 minutes. Remove the foil and cook an additional 5 minutes, or until the cheese has melted and begun to brown. Remove them from the oven and let the sliders sit for 5 minutes before serving.