To make the burgers, mince the onion, shiitake mushrooms, and carrots, microwave for 2.5 minutes, then cool. Grate the onion for the sauce.
Combine the Step 1 mixture, egg, panko, salt, pepper, and nutmeg with the meat to make the hamburger mix.
Divide the hamburger mix into 4 portions, wrap each around a piece of cheese and form into patties. Make a slight depression in the center of each patty, taking care not to let the cheese fall out.
To grill the patties, heat oil in a frying pan, then add the burgers. Once browned on one side, flip over, cover the pan with a lid, then cook through. When both sides are browned, transfer to a serving dish.
Without cleaning the pan, lightly fry the grated onion. Add the ketchup, mirin, soy sauce, and Japanese Worcestershire sauce, then bring to a boil. Drizzle over the hamburgers, then serve.
Story Behind this Recipe
I added ketchup to my soy sauce-mirin onion sauce. I wanted to give it a western twist so that it would go better with these cheese-filled burgers.
It easier to wrap up the cheese if the hamburger mix is on the firm side. It also prevents them from breaking apart when grilled.