Make the patties. Combine ground beef with shredded carrots, grated onion, soy sauce, cornstarch, white pepper, and Chinese 5-spice powder in a large bowl. Mix until all the ingredients are fully incorporated. Shape patties into 4 equal patties and lay on a lined baking sheet. Transfer to the refridgerator and chill for 30 minutes.
Add rice flour to a shallow bowl. Carefully coat patties in flour, shaking off excess.
Heat oil in a nonstick skillet over medium-high heat. Add the patties to the pan and cook until browned, about 3-4 minutes. Flip over and cook on the other side until cooked and the juices run clear. Remove from heat and keep warm.
In the same pan, add the Canadian bacon and cook until lightly browned, about 2 minutes. Remove from heat and set aside.
Wipe pan clean with paper towel. Heat 1 tablespoon oil over medium-high heat. Crack eggs into the pan and fry until the desired degree of doneness. Remove from heat and set aside.
Assemble burgers by spreading kewpie mayo and ketchup on both the top and bottom buns. Place lettuce on the bottom bun, followed by the tomatoes, cucumber, beef patty, bacon, and fried egg. Top with remaining bun and serve immediately.