In a medium skillet, saute onions with the butter over medium-low heat until the onions are golden brown. Add green enchilada sauce and green chilies to onion mixture and heat until very warm.
Lightly butter one side of each tortilla. Heat, butter side down, in a non-stick skillet over medium heat. Sprinkle cheese over each tortilla. Remove from skillet once cheese is melted and tortilla is lightly toasted. Repeat with remaining tortillas.
Beat eggs in a medium bowl and salt and pepper to taste. Melt a tablespoon of butter in non-stick skillet (the same one you used to heat the tortillas) and add the eggs. Stir until the eggs are set and remove from heat.
To serve, stack two cheesy tortillas on top of one another. Cover with green chile sauce and eggs. Top with your favorite toppings, such as salsa, sour cream, pico de gallo or guacamole/avocado slices.