Preheat the oven to 450°F (230°C).
Add sweet potato wedges to a large bowl. Drizzle with olive oil and sprinkle over the smoked paprika, garlic powder, onion powder, salt, and pepper. Toss until well coated.
Spread the wedges onto a large baking sheet in a single layer, leaving some space between them.
Roast for 25-30 minutes, turning once halfway through cooking. The wedges should be soft in the center with nicely browned edges. Pop under the broiler for the last 2-3 minutes for more browning, if desired.
In a large skillet, heat oil over medium-high heat.
Add onion and cook for 3-4 minutes until softened. Add beef and cook until browned, breaking it into larger crumbles as it cooks.
Stir in Worcestershire sauce, mustard, garlic powder, salt, and pepper. Reduce the heat to low and stir in the American cheese until melted.
In a small bowl, whisk together mayo, ketchup, mustard, pickle brine, garlic powder, and smoked paprika.
Divide lettuce, sweet potato wedges, and beef between bowls. Add the pickles, pepperoncinis, and pickled red cabbage around the sides.
Drizzle with the burger sauce and serve.