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 cabbage mixture; cook and form it into a rectangle. Split the rectangle into square patties; continue to cook until golden brown, 1 ½ to 2 minutes per side.

Stack patties on top of one another onto one slice of toast 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 patties; top with ketchup, mayonnaise, and the remaining slice of toast. Cut in half and serve.

Into a cocktail shaker, pour in the water, matcha, and honey. Close lid and shake vigorously for 10-20 seconds.

Fill the glass with ice. Pour the matcha mixture into the glass.