In a large bowl, whisk together the flour, cornstarch, baking powder, salt, and cold water until smooth. Whisk in the egg, if using.
Add the zucchini, carrot, onion, green onions, mushrooms, and bell pepper. Stir until all vegetables are evenly coated in batter.
Heat 1–2 tablespoons of oil in a large nonstick skillet over medium-high heat until shimmering.
Scoop half the vegetable mixture into the pan and spread into a thin, even round (about 1/4-inch thick). Cook 3–4 minutes until the bottom is golden and crisp.
Flip carefully and cook the second side 3–4 minutes until crisp and cooked through, pressing gently with a spatula for even browning. Transfer to a paper towel–lined plate or cutting board. Repeat with remaining mixture, adding more oil if the pan looks dry.
Make the dipping sauce: stir together soy sauce, rice vinegar, sesame seeds, sesame oil, sugar, minced garlic, and red pepper flakes (if using).
Slice pancakes into wedges or squares and serve warm with dipping sauce.