Cook's Note: If you have access to Chinese chives or ramps when in season, replace half of the scallions with these.
For the dipping sauce: Combine the soy sauce, 2 tablespoons water, the gochugaru, rice vinegar, sesame seeds, scallions and garlic in a small bowl and set aside.
For the pancakes: Heat 1 tablespoon sesame oil in a large nonstick skillet over high heat. Add the mushrooms, garlic and zucchini and sprinkle with salt and pepper. Spread the vegetables in the skillet and cook without mixing until they turn brown, for 1 to 2 minutes. Using a wooden spoon, toss the vegetables and cook until brown and dry, another 2 to 3 minutes. Transfer to a bowl and allow to cool.
Put the kimchi in a paper towel, squeeze out the excess liquid and roughly chop. Add the kimchi to a large bowl with the pork, scallions, carrots, chiles, shallots and the zucchini and mushroom mixture. Add the flours, eggs, 1 teaspoon salt, the remaining 2 tablespoons sesame oil and 1 1/4 to 1 1/2 cups cold water and mix until all of the ingredients are incorporated. The batter will be thick but loose.
Heat 2 tablespoons canola oil in the same skillet over medium-high heat. Add 3/4 cup of the batter to the skillet and spread it evenly until 1/2-inch thick, using a spatula if necessary. Lower the heat to medium and cook until the sides are crisp and small bubbles form on top, for 3 to 4 minutes. Flip the pancake and cook the other side until brown and crispy, for 2 to 3 minutes. Flip the pancake two more times, once again on each side, and cook for 30 seconds per side to ensure the pancake becomes crispy. Repeat with the remaining pancake batter.
Transfer the pancakes to a serving platter and cut into wedges. Serve with the dipping sauce.