Preheat oven to 350 degrees F. Spray an 8x8-inch baking pan with cooking spray.

Prepare sausage patties. Combine ground pork with onion, carrot, soy sauce, cornstarch, salt, white pepper, and Chinese 5 spice in a large bowl. Mix and shape into 4 equal patties. Place on a baking sheet lined with parchment paper. Bake sausage patties for 10 minutes.

Add eggs to a large bowl and whisk to break up the yolks. Add milk and whisk until well combined. Add scallions and salt and whisk until combined. Pour egg mixture into prepared baking pan. Bake eggs for 15 minutes. Remove and cut baked eggs into quarters. Keep warm.

Heat 1 tablespoon oil in a large saute pan over medium heat. Lay sausage patties in one layer and cook until browned, about 2 minutes. Flip and sear on the other side. Remove from heat and keep warm.

Assemble sandwiches. Smear kewpie mayo on the bottom of an English muffin and top with sausage patty, egg square, and cheddar. Place the remaining muffin on top and serve immediately.