In a small saucepan, melt the unsalted butter over medium heat. Once melted, add the flour and cook for 1-2 minutes, stirring constantly, to form a roux.

Gradually add the water, whisking continuously to prevent lumps.

Stir in the brown sugar, soy sauce, vinegar, and ketchup. Continue to cook, stirring frequently, until the sauce thickens and becomes smooth. Set aside.

In a bowl, beat the eggs with a pinch of salt and pepper. Place the bread crumbs in a separate bowl.

Heat a generous amount of cooking oil in a frying pan over medium heat.

Dip each egg into the bread crumbs, ensuring it is fully coated. Then, dip it into the beaten eggs and coat again with the bread crumbs.

Fry the coated egg cutlets in the hot oil until they are golden brown and crispy on both sides, about 2-3 minutes per side. Remove from the pan and drain on paper towels.

In a small bowl, mix together the mayonnaise, honey, and lemon juice until smooth. Adjust the sweetness or acidity to taste.

Butter the bread slices and toast them in a pan or toaster until golden and crispy.

Spread the mayonnaise sauce evenly over the toasted bread slices.

Layer the cheddar cheese on top of the mayonnaise sauce.

Add the pickle cucumber slices and sliced cabbage on top.

Drizzle 1.5 tablespoons of the egg cutlets sauce over the assembled toast.

Place the fried egg cutlets on a serving plate. Serve alongside the cabbage toast, optionally garnished with extra egg cutlets sauce if desired.