Preheat the barbecue to medium-high heat (approximately 220°C – 425°F).
In a cast iron skillet, mix the chopped red pepper with the red onion, mushrooms, pepperoni slices, thyme, and oil. Season with salt and pepper.
Place the frozen duck burgers and the cast iron skillet on the hot and oiled barbecue grill. Close the lid and cook for 10 to 14 minutes, turning the burgers and stirring the vegetables halfway through cooking, until the internal temperature of the burgers reaches 74°C (165°F) on a cooking thermometer.
Meanwhile, pour the marinara sauce into a small saucepan. Bring it to a boil.
Toast the hamburger buns for 15 seconds on each side on the hot barbecue grill.
Spread marinara sauce on the bun bottom and top with the vegetable mixture, duck burgers, and cheese. Cover with the bun tops.
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