Preheat oven to 350˚F. In a small bowl, combine the sugar, soy sauce, rice vinegar, sake, lime juice, lime zest and ¼ tsp. salt. Stir until the sugar is dissolved; set the sauce aside.
Add oil to a large frying pan over medium high heat. Add the pork, seasoned with ¾ tsp. salt and cook until just done, about 3 to 4 minutes. Drain and discard the oil, then place the pork into a large mixing bowl.
Add the onions to the same frying pan and cook over medium heat until translucent, 8 to 10 minutes. Add the garlic, mushrooms and bamboo shoots; cook for 1 to 2 minutes more; remove from heat and add to the pork. Add the sauce to the pork mixture and stir to combine well.
Place the tofu cubes in the bottom of a medium casserole dish. Pour the pork mixture on top of the tofu; spread evenly. Pour the beaten eggs over the top to cover.
Bake for 30 minutes or until set. Portion the casserole into 4 servings; garnish as desired. Enjoy this easy-to-prepare Autumn dish with an ice-cold Asahi beer.