Add sliced carrots and cabbage to the bowl with the eggs and mix well.
Add the oil to a pre-heated pan and add the egg mixture. Using a spatula, divide the mixture in two and work to form each half into the shape of the bread slices. When firm enough to hold, flip each half over to cook on the other side - about 2 - 21/2 minutes per side. Remove when done and keep warm.
Melt 1 tbsp butter in the pan. Add two slices of bread to the melted butter and toast the bread until lightly browned, flipping the bread to toast the other side. Remove to a plate and sprinkle a little sugar on the warm toasted bread. Repeat with the remaining two slices of bread.
To assemble the street toast sandwiches, top one slice of toasted bread with one of the cooked egg sections. Layer one half of the sliced cucumber on top of the egg. Squeeze some ketchup and mayonnaise on top of the cucumber, and top with a slice of toasted bread, Repeat to make the second toasted sandwich.