Place the block of tofu onto a plate and place another plate on top. Set a 3 to 5 pound weight on top (a container filled with water works well). Press the tofu for 10 to 15 minutes, then drain off and discard the accumulated liquid.
Cut the tofu into 1/2-inch cubes.
Combine the tofu, ground beef, shiitake mushrooms, miso paste, egg, salt, pepper, and nutmeg in a large bowl. Divide into 6 balls, and flatten into patties.
Stir together the mirin, soy sauce, garlic paste, and ginger in a small bowl. Set aside.
Heat the vegetable oil in a large skillet over medium-high heat. Brown the burgers on each side, about 2 minutes per side. Reduce the heat to low; cover the pan and cook until the juices run clear, about 5 minutes.
Drain and discard any excess grease.
Pour the soy sauce mixture into the pan. Keep moving the pan so the sauce doesn't burn, and flip the burgers occasionally to coat both sides with the sauce. The sauce will thicken and create a shiny glaze on the burgers. When all the sauce has been used up, the burgers are ready.
