Boil vinegar, sugar and water in a saucepan. Add the red onion and let the mixture cool in the refrigerator.
Cut the carrots into very thin strips with a peeler. Heat oil in a saucepan. Test the temperature with a small piece of bread - bread should be golden. Fry the carrots in the hot oil for about 3 min. Remove them with a slotted spoon and let them drain. Lay the carrot fries on a paper towel and sprinkle with sea salt.
Combine the mince with salt and freshly ground black pepper and form 4 burgers about 10 cm in diameter. Melt the butter in a sauté pan until golden. Fry the burgers for about one and a half minutes on each side over a moderate heat for about 2 min. Cover the pan with a lid, remove it from the heat and let the burgers rest. Warm the buns in the oven and cut them in half. Spread them with chili mayonnaise and add salad and sprouts. Lay the cheese and the pickled red onions on the burgers. Add the top bun and serve with carrot fries.