Sandwich 12–14 oz. extra-firm tofu between several layers of clean kitchen towels and gently press to expel excess liquid. Pat dry, then cut into 1" cubes.

Heat 2 Tbsp. vegetable oil in a large nonstick skillet over medium-high. Cook tofu, stirring and turning occasionally, until golden brown all over, 6–8 minutes. Generously season with kosher salt, then sprinkle 2 tsp. sugar evenly over. Cook, stirring constantly, until tofu is starting to brown, about 1 minute. Using tongs or a slotted spoon, transfer tofu to a plate, leaving behind as much oil as possible.

Heat remaining 2 Tbsp. vegetable oil in same pan. Add 1 lb. Japanese or Italian eggplants (about 3), cut in half lengthwise, cut crosswise ½" thick, season with salt, and cook, stirring often, until brown and starting to soften, 5–7 minutes. Add 4 large scallions, white and pale green parts thinly sliced, 4 garlic cloves, finely chopped, one 1" piece ginger, peeled, finely chopped, and ¼ tsp. crushed red pepper flakes and cook, stirring, until fragrant and starting to soften, about 1 minute. Add 2 Tbsp. soy sauce, remaining ½ tsp. sugar, and 2 Tbsp. water. Bring to a simmer (it might happen immediately), then reduce heat to low. Cover and cook until eggplant is starting to fall apart, about 2 minutes. Return tofu to pan; stir to coat. Remove from heat; stir in 1½ tsp. unseasoned rice vinegar and ½ tsp. fish sauce (if using).

Serve tofu mixture in a shallow bowl over steamed rice, topped with thinly sliced dark green scallion parts and drizzled with sesame oil.