Freeze the mozzarella string cheese sticks for at least 30 minutes — this is critical so they don't melt out of the batter during frying. While they freeze, make everything else.
Make the kimchi dust: Spread the drained, squeezed-dry kimchi in a thin layer on a parchment-lined microwave-safe plate. Microwave in 2-minute intervals, tossing between rounds, for about 6-8 minutes total until completely dried out and crispy. Let cool, then crush into a coarse powder with your fingers. Toss with 1 tsp sugar. Set aside.
Make the batter: Whisk together flour, sugar, instant yeast, salt, and baking powder in a large bowl. Add warm milk and egg. Stir until a thick, sticky batter forms — it should coat the back of a spoon heavily. Cover with a towel and let rest for 15 minutes at room temperature. The batter will puff slightly.
Make the spicy honey mustard: Stir together yellow mustard, honey, gochugaru, and rice vinegar in a small bowl. Set aside.
Assemble the corn dogs: Insert a wooden chopstick or skewer lengthwise through each hot dog, leaving 3 inches of stick exposed as a handle. Press a frozen mozzarella stick alongside the hot dog and push the skewer through both to hold them together. Pat the outside completely dry — batter will not stick to moisture.
Pour panko breadcrumbs into a shallow dish. Pour the batter into a tall drinking glass or jar for easier dipping. Dip each assembled corn dog into the batter, turning to coat evenly. Let excess drip off for 5 seconds, then immediately roll in panko, pressing gently so the crumbs adhere on all sides.
Heat 3-4 cups of neutral oil in a deep heavy pot or Dutch oven to 350°F. Use a thermometer — temperature matters here. Fry 2 corn dogs at a time for 3-4 minutes, turning occasionally with tongs, until deep golden brown and the batter is cooked through. Drain on a wire rack set over a sheet pan.
Immediately roll each hot corn dog in granulated sugar while still glistening with oil — the sugar sticks to the surface and creates that signature sweet-savory Korean corn dog coating.
Drizzle generously with spicy honey mustard, sprinkle with kimchi dust, and serve immediately on a piece of wax paper. Eat standing up — this is street food, not a sit-down situation.