Cut 4 lengthwise slices from block of tofu, each about 1/2 in. thick. Transfer to a 9-in. square baking pan.
Grate ginger with a Microplane, then press solids through a fine-mesh sieve into a bowl to get juice. Measure 2 tsp. ginger juice into another bowl. Whisk in sugar, pepper, both kinds of soy sauce, sesame oil, and 1 tbsp. canola oil, then pour over tofu. Turn tofu to coat and set aside, uncovered, 1 to 2 hours, turning halfway through.
Brush a cast-iron grill pan lightly with more canola oil and heat over medium-high heat until pan is hot enough to evaporate a bead of water. Sear tofu, 2 pieces at a time (reserving marinade), until grill marks appear and tofu releases from pan easily, 3 minutes per side. Transfer to a plate.
Toast rolls or bread, cut sides up, in a 300° oven until crisp, about 15 minutes. Meanwhile, toss zucchini in leftover tofu marinade and grill until tender and browned, about 4 minutes per side. Save remaining marinade.
Spread about 1 tsp. mayonnaise on each half of roll, then drizzle about 1/2 tsp. of reserved marinade and 1/2 tsp. Sriracha on each. Sprinkle half of cilantro on bottom roll halves. Add tofu, bacon, zucchini, and avocado. Top with remaining cilantro and tops of rolls.