Press tofu: Wrap tofu in a clean kitchen towel or paper towels. Place on a plate and place a 1-2 lb object on top*. Leave for 20-30 minutes.
Slice eggplant: For large globe eggplants, cut slices about ¾" thick, then cut each slice into quarters. For thin Japanese eggplants, just cut into slices.
Salt eggplant: Place eggplant slices in a colander over the sink. Sprinkle with salt and let the eggplants sit for 20-30 minutes, during which time the eggplant slices should lose water and become softer.
Mix sauce: In a small bowl, mix together grated ginger, soy sauce, sake, mirin, sugar, and sesame oil. Set aside.
Pan fry tofu: Cut tofu into desired shape. Heat 1 tbsp of oil in a pan over medium heat. Add tofu and cook until golden brown and crisp on the bottom, then flip and fry the other side. Remove from pan.
Prepare eggplant: Rinse eggplant slices and pat dry. Place in a bowl and add cornstarch, mixing until well coated.
Pan fry eggplant: Heat 1 tbsp oil in the same pan used for the tofu. Add eggplant slices and cook until crispy and browned on both sides. Remove from pan.
Cook onion and garlic: Heat ½ tbsp oil in the same pan. Add onion and garlic and cook for a few minutes until onion is tender.
Combine: Add in the tofu and eggplant. Pour the sauce over and stir to coat. Cook for another minute until all the liquid has evaporated. Top with green onions, stir, and remove from heat.
*Try a container of rice or beans, a plate with a heavy book on top, a container of protein powder, etc. Try to choose objects that are flat on the bottom so they don't roll off.