Crack the eggs into a mixing bowl and whisk them thoroughly until the yolks and whites are well combined.
Stir in the sugar and continue mixing until fully dissolved into the eggs.
Place a small pan over medium heat and let the butter melt completely, ensuring it coats the surface evenly.
Pour the egg mixture into the pan and allow it to set slightly before gently folding the eggs from the edges toward the center.
Continue folding and stirring the eggs gently until they are softly scrambled and cooked to your preferred consistency, then remove them from the pan and set aside.
If the pan looks dry, add a little more butter and let it melt.
Place the bread slices in the pan and toast them on both sides until golden brown, pressing slightly to ensure even browning.
Lay one cheese slice on each piece of hot toasted bread to soften slightly from the residual heat.
Spoon the scrambled eggs onto one slice of bread, then cover with the other slice to form a sandwich.
Press lightly, cut if desired, and serve warm.
