Make the burgers: Pour about 1 tablespoon of olive oil into a frying pan set to medium heat. Add the onions and saute until translucent. Transfer to a bowl and cool.
In a bowl, combine the panko bread crumbs, milk and egg. Place the ground beef in a large bowl and add this mixture, the onions, salt and pepper. Knead with your hands until fully incorporated.
Divide the mixture into 4, and create patties, placing a square of cheese in the center of each patty.
Pour 1 tablespoon of olive oil into a pan set to medium heat. Once the oil has heated up, add the patties and cook for 2 minutes. Flip the patties over and place a lid on the pan. Cook for 8-10 minutes, until the patties are fully cooked through.
Serve in hamburger buns with your choice of toppings!