Wash and dry all produce. Preheat toaster oven or oven to 400 degrees. Core, seed, and dice tomato. Peel, halve, and thinly slice onion. Wrap garlic in aluminum foil, sealing to make a pouch, then roast in oven until very soft, about 20 minutes.

Heat a drizzle of oil in a large pan over medium heat. Add onion and cook, tossing, until softened, 4-5 minutes. Season with salt and pepper. Stir in tomato, 1 tsp sugar, and 1 TBSP vinegar (we’ll use the rest later). Cook, stirring, until soft and jammy, 4-5 minutes. Season with salt and pepper. Remove mixture from pan and set aside.

Divide beef in half and flatten each piece into a wide, roughly 1/2-inch-thick circle. Divide cheddar between the centers of each circle. Fold edges of meat around cheddar, shaping and sealing to create a cheese-stuffed patty. Season all over with salt and pepper.

Carefully wipe out pan used for onion jam with a paper towel, then add a drizzle of oil to it and heat over mediumhigh heat. Place beef patties in pan and cook to desired doneness, 3-6 minutes per side.

While burgers cook, halve buns and place on a baking sheet. Toast in oven or toaster oven until golden brown, 2-3 minutes. Once garlic is done roasting, mash with a fork until smooth. In a small bowl, combine 1/2 TBSP mayonnaise (we’ll use more in the last step), remaining vinegar, a large drizzle of oil, and as much garlic as you like. Season with salt and pepper.

Spread buns with ketchup and 1/2 TBSP mayo (you’ll have some left over). Divide beef patties between buns and top with onion jam and a small handful of arugula. Toss remaining arugula with dressing in a medium bowl and serve on the side.