Heat the olive oil in a large skillet over medium heat. Add the ground beef and cook, breaking it up with a spoon, until browned. Drain any excess grease.
Add the onion and garlic to the skillet. Cook until the onion is softened, about 5 minutes.
Add the beer to the skillet, scraping up any browned bits from the bottom of the pan. Bring to a simmer and cook for a few minutes until the beer reduces slightly.
Stir in the beef broth, ketchup, tomato paste, Worcestershire sauce, Dijon mustard, smoked paprika, cayenne pepper, salt, and black pepper. Bring to a simmer, then reduce heat to low and cook for 15-20 minutes, or until the sauce has thickened.
Remove the skillet from the heat and stir in the shredded cheddar cheese until melted and smooth.
Assemble the sandwiches: Divide the sloppy joe mixture among the toasted hamburger buns. Top each with 2 slices of bacon and enjoy!