Make the batter: In a large bowl, whisk together the flour, baking powder, salt, and sugar. Make a well in the center, add the egg and milk, and whisk until smooth. Transfer the batter to tall glass, then chill in the refrigerator until ready to use.
Make the potatoes: Dice the potatoes into ¼-inch cubes, then transfer to a large bowl and cover with cold water. Soak for 1 hour to remove excess starch, then drain.
Bring a medium pot of water to a boil. Add the potatoes and blanch for 3–4 minutes, until cooked ¾ way through. Drain and rinse with cold water. Pat the potatoes dry and transfer to a bowl. Sprinkle the flour over the potatoes and toss to coat.
Make the cheese dogs: Heat 4–5 inches of vegetable oil in a large pan over medium heat until the temperature reaches 350°F (180°C).
Cut each hot dog in half crosswise.
Cut the mozzarella block into 4 rectangles to match the size of the hot dog halves.
Thread a hot dog half and a cheese stick onto each wooden chopstick. Lightly coat in flour.
Transfer the potatoes to a plate or baking sheet. Add the panko bread crumbs to a separate plate or the same baking sheet.
Dip a cheese dog in the batter to coat fully, then place on the potatoes, packing tightly to adhere. Coat in the panko bread crumbs. Repeat with the remaining cheese dogs.
Fry the potato hot dogs, 2 at a time, in the hot oil for 5–7 minutes, turning occasionally, until golden brown on all sides. Transfer to a paper towel to drain.
Sprinkle the potato hot dogs with sugar and drizzle with ketchup.