Rinse the enoki mushrooms under cold running water, drain well, and pat dry with clean paper towels to remove excess moisture. Trim off about 1.5 inches from the tough base and cut the mushrooms into 1-inch pieces.
In a large mixing bowl, combine the enoki mushrooms, eggs, all-purpose flour, cornstarch, green onions, salt, and black pepper. Mix until the mushrooms are evenly coated and the batter is sticky.
Heat 2 tablespoon of vegetable oil in a 10-inch nonstick pan over medium heat, swirling the pan to evenly coat the bottom. Add half the batter and spread it evenly into a circle with a spatula. Fry for about 5-7 minutes until the bottom is golden brown and a firm crust forms.
Carefully flip the pancake with a large spatula. Alternatively, slide it onto a large plate, place the pan upside down over the plate, and carefully flip both together to return the pancake to the pan. Pour another 2 tablespoon of oil around the edges and fry for about 4-6 minutes until golden brown on the other side.
Transfer the cooked pancake to a plate. Repeat Steps 3-4 with the remaining batter and 4 tablespoon of oil.
Transfer both pancakes to a cutting board and slice into squares with a sharp knife, or cut them into smaller pieces using a pizza cutter or clean kitchen scissors.
In a small bowl, combine the soy sauce, rice vinegar, sugar, toasted sesame oil, and sesame seeds. Stir until the sugar dissolves. Serve with the warm pancakes.