Heat oven to 375°F. Whisk eggs, milk, salt, and pepper. Melt butter in skillet; gently scramble eggs over medium-low 4 to 5 minutes until barely set.

Fold ham into eggs. Warm tortillas in dry skillet 15 to 20 seconds per side until pliable; keep covered while working.

Spread 1/2 cup sauce in 9-by-13-inch dish. Divide filling among tortillas; roll and arrange seam-side down in two snug rows.

Cover with remaining sauce and Monterey Jack. Loosely foil-cover; bake 15 minutes.

Uncover; bake 10 minutes more until center filling 165°F and sauce bubbling. Rest 5 minutes; serve two enchiladas per person.