For the compound butter: Add the butter, cream cheese, pepper jack, dill, parsley, paprika, garlic, 2 tablespoons salt and 1 tablespoon pepper to a food processor and pulse, occasionally scraping down the sides, until well combined. Place the compound butter on a piece of plastic wrap and roll it in a tube shape about 2-inches in diameter, twisting the ends to secure. Place it in the freezer until completely solid, at least 3 hours.
Divide the chicken into four 8-ounce patties. Remove the log of compound butter from the freezer and place it in the center of the chicken. Wrap the chicken around the compound butter and seal to form it tightly and evenly around--like a large egg shape.
For the breading: Preheat the oven to 350 degrees F. Place a rack on top of a baking sheet.
Set up a dredging station with the flour, beaten eggs, seasoned Italian breadcrumbs and panko. Season each with 2 tablespoons salt and 1 tablespoon pepper. Dredge the chicken lightly in the flour, egg, then Italian breadcrumbs. Pass the chicken back through the egg and then finally in the panko for a second coating. Place on the prepared rack.
Fill a pot with 2 to 3 inches of oil and heat to 350 degrees F. Fry the chicken until golden brown, 2 to 3 minutes per side. Place the chicken back on the rack and bake until cooked through, 5 to 6 minutes. Let the chicken rest for 1 to 2 minutes before serving.