Preheat the oven to 220°C (200°C fan). Peel, halve, and slice the onions. In a medium-sized pot, cook the onion with 1Tbsp butter, 5Tbsp balsamic vinegar, 2Tbsp water, 3Tbsp sugar, and a large pinch of salt and pepper on low to medium heat until the onions are soft and jammy, 8–10min.
Cut the sweet potatoes into wedges, then add them to a parchment-lined baking tray and toss the wedges with 2Tbsp olive oil and ½tsp of salt. Bake in the oven for 20–25min., turning the wedges halfway through baking.
In a large bowl, mix 1Tbsp olive oil, 1Tbsp balsamic vinegar, 1tsp sugar, ½tsp mustard and a large pinch of salt and pepper to make a salad dressing. Set aside.
Peel and grate the garlic. In a bowl, mix the minced meat with the garlic, and a large pinch of salt and pepper. Shape the meat into 4 patties. Cut the buns open.
Heat a large pan on medium-high heat. Add 1Tbsp olive oil and the patties and fry for 3–4min. until browned on the first side. Then flip the patties, top with ½ of the cheese and cover with a lid to help the cheese melt. Fry for 3–4min. until the second side is browned and cooked to the desired doneness. Let rest covered on a plate.
Turn on the oven grill function at 220°C (200°C fan). Divide the fries over plates, and put the buns onto the tray. Sprinkle the rest of the cheese on the bottom buns and toast the buns for 2–3min. until golden and the cheese is melted. Toss the lettuce with the dressing. Add the patties to the buns and top with some onion jam as desired. Serve with the sweet potato fries and salad on the side.