Preheat the oven to 400°F. Line a large baking sheet with parchment paper.
Make the marinade by combining all ingredients in a wide bowl.
Slice the tofu into 12 pieces. Dip each piece of tofu into the marinade, then place on the baking sheet.
Bake for 40 minutes. Halfway through, baste each piece with the marinade, then flip and baste the other side.
To assemble each sandwich, place four pieces of tofu onto the bottom bun. Top with a slice of cheese, then place in a sauté pan over medium-low heat, covered, until the cheese melts. Top with some crispy onions, extra sauce, and the top bun. Enjoy!