Melt butter in a large nonstick skillet over medium heat; add eggs. Cook, without stirring, until eggs begin to set on bottom.
Draw a spatula across bottom of skillet to form large curds. Continue cooking until eggs are thickened but still moist (do not stir constantly). Remove eggs from heat.
Add half of cheese to eggs, stirring until cheese melts; chill remaining cheese. Stir ham into egg mixture.
Spoon mixture evenly down centers of tortillas; roll up, jellyroll fashion. Place seam side down in a lightly greased 13- x 9-inch baking dish. Cover and chill 8 hours. Let stand at room temperature 30 minutes.
Bake at 350º for 40 minutes or until thoroughly heated.
Combine remaining cheese and milk in a small saucepan; cook over medium heat, stirring until cheese melts. Spoon over enchiladas.