Squeeze the chopped kimchi over a bowl, catching the liquid separately, and set both aside. You want the kimchi damp, not dripping.
Whisk the flour, potato starch, egg (if using), sugar, and the saved kimchi brine together, then add cold water a little at a time until the batter runs off a spoon in a steady, thin ribbon. Juicier kimchi means less added water; drier kimchi means more.
Fold in the squeezed kimchi and half the scallions. Don't overmix — a few uneven streaks are fine.
Heat 2 tablespoons of oil in a nonstick pan over medium-high heat until it shimmers.
Pour in a thin layer of batter, about 5 mm / 1/4 in deep, tilting the pan to spread it. Thick pancakes steam in the center before the outside browns.
Cook undisturbed for 3 to 4 minutes, until the edges visibly darken and lift slightly off the pan.
Add a teaspoon of oil around the rim, then flip with a wide spatula in one motion.
Press down once with the spatula and cook the second side 2 to 3 minutes, until it's deep red-brown and firm all the way through, not pale in the middle.
Slide onto a cutting board, cut into wedges, and serve with the dipping sauce made from soy sauce, rice vinegar, gochugaru, and sesame seeds.