In a non-stick skillet, brown tofu in oil. Add ginger and hoisin sauce. Let cool.
Spread about 125 ml (1/2 cup) of rice on the matte side of the nori, up to 2-cm (3/4-inch) from one edge of the narrow side, working with moistened fingers. Spread about 30 ml (2 tablespoons) of sesame seeds in the centre of the sheet. Lightly moisten countertop. Flip sheet, rice side down.
Place tofu on the upper third of the sheet. Add some avocado, carrot and cucumber julienne. Roll, starting with the rice-less side of the nori. Compress roll with a bamboo mat, applying uniform pressure. Dip the tip of a knife in water and cut roll into ten bites. Place on a serving plate. Repeat with remaining ingredients.
Cut cheese into slices the width of the sushi. In a non-stick skillet, grill slices on both sides. Let cool slightly. Place on sushi and serve.
This recipe was developed in collaboration with Genevieve Everell.