Beat eggs, milk/water, salt and pepper in bowl until blended.
Heat 1 tsp butter in nonstick skillet over medium heat until hot. Pour in egg mixture. As eggs begin to set, gently pull the eggs across the pan with an inverted turner, forming large soft curds. Continue cooking – pulling, lifting and folding eggs – until thickened and no visible liquid egg remains. Do not stir constantly. Remove from pan and clean skillet.
Spread remaining 2 tsp butter evenly on one side of each bread slice. Bread 2 slices in skillet, buttered side down. Top evenly with scrambled eggs, cheese. Cover with remaining bread, buttered side up.
Grill sandwiches over medium heat, turning once, until bread is toasted and cheese is melted, 3 to 5 minutes.