In small bowl, whisk eggs. In large nonstick skillet, melt 1 tablespoon butter over medium-low heat. Add eggs and cook 4 minutes or to desired doneness, stirring occasionally to scramble. Transfer eggs to cutting board; divide into 4 equal pieces.
In same skillet, cook ham over medium heat 3 minutes or until golden brown and edges are crisp, turning once.
Spread mayonnaise on both sides of toast. Place 4 pieces of toast on work surface; top with half the cheese, ham, eggs, kimchi, and remaining cheese and 4 pieces of toast.
In 2 batches, in same skillet, add sandwiches; cook 6 minutes over medium heat or until bread is golden brown and cheese is melted, turning once and pressing down occasionally with large spatula. Cut sandwiches diagonally in half. Makes 4 sandwiches.
