Adjust rack to top position and preheat oven to 425 degrees. Wash and dry all produce. Cut broccoli florets into 1-inch pieces. Toss on a baking sheet with a drizzle of oil, salt, and pepper. Place garlic in the center of a small piece of foil; cinch into a packet and place on same sheet. Roast until broccoli is tender and garlic is softened, 15-20 minutes.

Meanwhile, halve, peel, and thinly slice onion. Dice tomato. Heat a drizzle of oil in a large pan over medium heat. Add onion and cook, stirring, until softened, 4-5 minutes. Season with salt and pepper. Stir in tomato, 1 tsp sugar (2 tsp for 4 servings), and half the vinegar (all the vinegar for 4). Cook, stirring, until soft and jammy, 4-5 minutes. Season with salt and pepper. Turn off heat; remove jam from pan and set aside. Wipe out pan.

In a medium bowl, combine beef, half the Fry Seasoning (all for 4 servings), 1/2 tsp salt (1 tsp for 4), and pepper. Form mixture into two wide, roughly 1/2-inch-thick rounds (four rounds for 4). Divide cheddar between the centers of each round. Fold edges of meat around cheddar, shaping and sealing to create cheese-stuffed patties.

Heat a drizzle of oil in pan used for jam over medium-high heat. Add patties and cook to desired doneness, 3-5 minutes per side.

While patties cook, halve buns; toast until golden brown. Once garlic is roasted, remove from foil and mash with a fork until smooth. In a small bowl, combine mayonnaise, sour cream, and mashed garlic. Season with salt and pepper.

Spread buns with ketchup and a thin layer of sauce. Fill with patties and top with tomato onion jam. Divide burgers and broccoli between plates. Serve with any remaining sauce on the side for dipping.