Preheat the grill, setting the burners on high. Oil the grate.
Slice the block of tofu into two rectangles lengthwise. Slice each rectangle into 12 square slices.
Stack three slices of tofu, inserting two slices of cheese in between. Insert two toothpicks at a slight angle to hold them together. Oil lightly. Repeat with the remaining slices. Season with salt and pepper.
Grill the tofu blocks for about 2 minutes on each side or until the cheese melts and begins to flow. Set aside on a warm plate. Remove the toothpicks.
Toast the buns. Spread the inside with mayonnaise and mustard. Place the tofu cakes on the bottom buns. Top with arugula. Cover with the top buns.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.