Combine the hard cider/beer, broth, and bratwurst in the Big Green Egg Dutch oven, bring to a simmer, and gently simmer for 15 minutes.
Meanwhile, heat the oil and butter on the Big Green Egg Half-Moon Cast Iron Plancha. Add the onions, sprinkle with salt and pepper, and cook until golden brown. Keep the onion mixture warm while you cook the bratwurst.
Transfer the bratwurst directly to the grill grate and cook, turning occasionally, until browned, about 10 minutes. Lightly grill the pretzel rolls cut side down. Serve the bratwurst topped with the onions and with the pretzel rolls and mustard on the side.