Slow cook the pork. Add the beef broth, onion, salt, pepper, garlic powder, mesquite seasoning and pork roast to a slow cooker. Cook on low for 6 to 8 hours, until the meat is fork-tender.
Add the barbecue sauce during the last hour of cooking.
Shred and drain. Remove the pork, shred it with two forks and let any excess juices drain off so the sandwiches stay crisp.
Prep the bread. Spread mayo on the outside of each slice of bread.
Build the sandwiches. Layer mozzarella, shredded pork and cheddar between the bread, mayo side out.
Toast. In a large non-stick skillet over medium-low heat, toast the sandwiches in batches 2 to 3 minutes per side, until golden brown and the cheese is melted.
Serve with sliced pickles on the side and french fries.
Notes
Leftover pork keeps up to 4 days refrigerated, so you can make sandwiches all week. A panini press works well too.