Whisk the buttermilk, beer, 1/2 teaspoon of salt and 1/2 teaspoon of pepper together in a large bowl. Add the chicken to the marinade, turn to coat, cover with plastic wrap and refrigerate for at least 2 hours or overnight.
Heat the oil in a large, deep skillet over medium heat until it reaches 350 degrees. While the oil is heating, whisk the eggs together in a medium bowl. Whisk the flour, the remaining 1 teaspoon of salt, and the remaining 1/2 teaspoon of pepper together in a second bowl, and mix the panko and parmesan together in a third bowl.
Dredge a breast through the seasoned flour, making sure both sides are evenly coated. Tap off the excess. Dip the floured breast in the egg mixture, and then press both sides into the panko/parmesan mixture.
Place the breaded chicken breast on a baking sheet or plate while you coat the remaining breasts. Place the chicken breasts in the skillet and fry on both sides until golden and crisp, about 4 minutes. Use tongs to transfer the chicken to a paper towel-lined plate to drain and cool completely.
Set a piece of fried chicken on the cutting board. Use a 1 1/4-inch round cookie cutter to stamp 24 circles from the chicken breasts. Spread 1/8 teaspoon of sauce on one side of each circle and cover the sauce with a piece of cheese. To serve, heat the oven to 350 degrees.
Place the chicken in the oven and bake until the cheese begins to melt, 2 to 3 minutes.
Remove from the oven and put on lollipop stick or wooden skewer. Serve immediately.