In a bowl, mix cabbage, green onion, salt, pepper, and cayenne until the mixture is soft, about 30 seconds. Mix in eggs with a fork until thoroughly combined.
Melt ⅓ of your divided butter in a nonstick skillet over medium heat. Add bread slices and flip a few times to make sure both sides are coated with butter. Grill bread until beautifully toasted, golden brown, and a little crispy on both sides. Transfer to a plate.
Melt the remaining butter in the same skillet over medium heat. Add the cabbage/egg mixture; cook until the egg set and starts to become firm then flip over and continue to cook until golden brown, 1 ½ to 2 minutes per side.
Portion to fit the bread and place onto the bottom bread slice and sprinkle with sugar.
Place ham slices into the hot skillet; cook over medium heat until sizzling, 1 to 2 minutes. Flip and top with cheddar slice; cook until melted, about 1 minute.
Place ham and cheese on top of the egg; top with ketchup, mayonnaise, and the remaining slice of bread. Cut in half and serve.
In a blender, combine the fresh or frozen strawberries, ripe banana, milk, and honey (if using).
If you prefer a thicker smoothie, add a handful of ice cubes to the blender.
Blend all the ingredients until smooth and creamy.
Taste the smoothie and adjust the sweetness by adding more honey if needed.
Pour the strawberry banana smoothie into glasses.
Serve the smoothies alongside the Korean Street Toast for a delicious and satisfying meal. Enjoy your meal!