For the tenderloin: Cut the tenderloin in 4 medallions. Butterfly each medallion crosswise, and then place each medallion between 2 pieces of plastic wrap. Using a meat mallet, pound until each patty is 1/8-inch thick and 6 to 8 inches wide.
Mix together the buttermilk, honey, granulated garlic, salt, onion powder and paprika in a bowl to make the marinade. Place the pork into a 9- by 13-inch dish and cover in the marinade. Marinate in the fridge for 1 to 2 hours.
In a large cast-iron skillet, heat oil over medium heat to 350 degrees F. Preheat the oven to 200 degrees F.
Set up a three-stage breading station, with the flour, eggs and breadcrumbs in three separate baking dishes. Place the marinated tenderloins first in the flour, shake off any excess, and then in the eggs, and let any excess drip off, and then in the breadcrumbs. Press the patties into the breadcrumbs until total coverage is achieved.
One at a time, fry each patty until golden brown, about 3 minutes, flipping if necessary. The pork is so thin, that by the time it is golden brown, it is done. Once the pork is finished, place it onto a wire-rack-lined baking sheet and place in the oven to keep warm, if you like. If not, no worries, it's just as good at room temperature.
For the apricot mustard spread: Mix the mayonnaise, mustard, apricot preserves, honey and horseradish in a bowl. Season with salt and pepper.
For the sandwich build: On each bottom bun, place 2 slices American cheese and then top with the warm pork tenderloin. Top the pork with red onions, pickles and lettuce. Schmear the apricot mustard spread onto the top bun and then top the sandwich.