Giardiniera butter: combine the softened butter with the chopped and minced giardiniera until everything is incorporated. Store in the fridge until about 30 minutes before sandwich time. At that point, remove it from the fridge to give the butter a chance to soften up.

Breaded Parm chicken cutlets: first, you need to butterfly your chicken breast and then cut it into two pieces. Butterflying cuts the chicken breast parallel to your cutting board so that you will have a flat, thinner piece of chicken. Here's a short butterflying tutorial if you've never butterflied a chicken breast before. After you've butterflied the chicken, go ahead and cut the large chicken piece in half so that you have two thin cutlets. 

Cover the cutlets with plastic wrap or, even better, a large zip-top bag that you have laid on top of the chicken. Using a meat mallet or a heavy pot, pound the chicken through the plastic to attempt to flatten each piece of chicken until it is around 1/2-inch thick. 

Set up your chicken dredging station. You will need three bowls or large plates for this. 

The first bowl will contain 1/4 cup all-purpose flour. Add 1/2 teaspoon salt and 1/2 teaspoon black pepper to the flour. The second bowl will contain 1 whole egg, whisked well. The third bowl will have 1/2 cup Panko breadcrumbs, 1/4 cup grated Parmesan cheese, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Whisk to combine. 

Add peanut or vegetable oil to a large pan or pot until it's round 1/2 inch deep. Place the pan or pot over medium-high heat and allow it to warm up until it's around 350 F (175 C).

Once the oil is up to temperature, carefully place each of your cutlets into the oil and fry them for 2 minutes, then flip and fry for an additional 2 minutes. The cutlets should be very thin so they will cook quickly. 

When the cutlets are fully cooked, remove them from the oil to a paper-towel-lined sheet pan or onto a cooling rack over a paper-towel-lined sheet pan. 

Broil buns and melt cheese: spread the giardiniera butter on both inner sides of your two split sandwich buns. Place on a foil-lined sheet pan, butter side up. Place both cutlets on the foil and top them with slices of cheese.

Broil for 3 to 4 minutes or until the buns are toasty, the butter is melted, and the cheese has also melted a bit. Remove and allow everything to cool for 3 or 4 minutes. The chicken should be very hot. 

Sandwich assembly: place a cutlet on top of the bottom bun. Spoon giardiniera on top of the melted cheese and then close the sandwich with the top bun. 

Serve and enjoy.