Barbecue onions: Cut the onions in half vertically. Cut off both ends and peel away the skin. With the middle of the onion face down, cut the onion into ¾ inch wedges. Heat 1½ tablespoons of the olive oil in a pan over high heat. Add the onions (you should hear them sizzle). After a minute or two, lower the heat to medium-low and cook the onions until brown and caramelized, about 10-12 minutes. Stir in the barbecue seasoning, soy sauce, and Worcestershire sauce. Let the flavors blend over low heat for about 10 more minutes. (At this point, the onions are ready to serve. However, you can let them sit longer if necessary. If they start to lose their shape, turn off the heat. You can reheat them over medium heat when everything else is ready.)
Filet patties: Preheat the oven to 375°F. Form the ground filet mignon into patties, 3½ inches in diameter and ¾ inches thick (about 3.3 ounce portions). Season the patties with salt and pepper. Heat the rest of the olive oil on a flat top grill over high heat. Sear the patties until brown and slightly crisp, about 2 minutes per side. Transfer them to the oven and cook for 6-8 minutes, or until desired doneness is reached.
Horseradish mayo: Meanwhile, whisk together the mustard, horseradish, and mayonnaise. Reserve in the fridge until ready to serve.
To serve: When the patties have cooked, place a piece of cheddar on each patty and broil for 30 seconds to melt the cheese. Transfer the patties to the pretzel buns, pile high with barbecue onions, and slather on the horseradish mayo.