In a saucepan, combine the soy sauce, sugar, and vinegar. Cook the marinade over low heat until thickened, 5 to 7 minutes.
Cut the tofu into four 1-inch-thick burgers. Transfer to a bowl, pour the marinade over the top, and let sit 15 minutes, turning once.
Place the burgers on a medium-hot grill and cook 1 to 2 minutes per side. (Or heat the oil in a grill pan over medium heat and cook the tofu 2 minutes per side.)
Top each burger with some of the Cheddar and cook 2 minutes more or until the cheese melts.
Top with some sprouts and sliced radishes. Serve on the rolls with mustard, if desired.