Mix the onion, garlic, salt, and sugar in a bowl or jar and pour vinegar over them until almost covered. Add a little water so they are covered by about an inch. Cover and store on the counter for an hour or two, or in the fridge overnight until ready to use.
Cut the watermelon into 2 rounds, the same size as your buns. Cut the feta brick in half the long way so it can be placed on top of the burgers.
Preheat grill or pan to medium heat.
Toast the buns until golden brown. Drizzle some oil and balsamic vinegar onto the buns.
Lightly season the burgers with salt and pepper. Grill the burgers on the first side, about 5 minutes, until the juices are collecting on top of the burger. Flip the burger and top with the feta and a little chili flake. Cover and cook another 5 minutes until the burgers are cooked through and the feta is soft.
Build your burger starting with the bottom bun, then the burger and feta, then the watermelon, then the onions and mint. Add the top bun and serve.