Heat the oven to 375°F. Whisk the eggs, milk, salt and pepper. Melt the butter in a skillet and gently scramble the eggs over medium-low heat for 4 to 5 minutes, stopping when barely set.
Fold the ham into the eggs. Warm the tortillas in a dry skillet for 15 to 20 seconds per side until pliable, keeping them covered as you work.
Spread 1/2 cup sauce in a 9-by-13-inch dish. Divide the egg filling among the tortillas, roll them and place seam-side down in two snug rows.
Pour over the remaining sauce and sprinkle with Monterey Jack. Cover loosely with foil and bake for 15 minutes.
Uncover and bake for 10 minutes more, until the middle filling reaches 165°F and the sauce bubbles around the rolls. Rest for 5 minutes and serve two enchiladas per person.