18 large eggs
1 cup whole milk
1 tablespoon unsalted butter
12 (10-inch) flour tortillas
1/4 cup Dijon mustard (optional)
24 slices (about 1 pound) thinly sliced deli ham
3 cup shredded Gruyère cheese
Whisk the eggs in a large bowl until the whites and yolks are completely mixed and the eggs are a bit frothy. Whisk in the milk until just combined.
Heat the butter in a large nonstick skillet over medium-low heat until melted. Pour in the egg mixture. Let sit undisturbed until the eggs just start to set around the edges, about 2 minutes. Using a rubber spatula, push the set eggs from the edges into the center. Spread the uncooked eggs back into an even layer. Repeat, pushing the set eggs from the edges into the center every 30 seconds until almost set, for a total cooking time of 6 to 8 minutes. (The top of the eggs should still be slightly wet.) Transfer the eggs to a bowl.
Wipe out the pan to reuse, or use a griddle and heat over medium heat. Meanwhile, assemble the quesadillas.
Stack the tortillas on a microwave-safe plate and cover with a damp paper towel. Microwave until soft and pliable, about 30 seconds.
Assemble the quesadillas: Place the tortillas on a work surface. If using, spread about 1 teaspoon of the mustard over the center of each tortilla. Top each tortilla with one slice of ham. Divide the eggs among the tortillas, spreading into an even layer but leaving a 1 1/2-inch border. Sprinkle the cheese over top of the eggs, then top each with one more slice of ham.
To fold the quesadillas, fold the top of the tortilla down over the filling to the center. Holding that piece down and working clockwise, continue folding the rest of the tortilla towards the middle until the filling is completely covered (you will have about 5 folds). Carefully flip the quesadilla over and repeat with the remaining tortillas.
Place the quesadillas, folded-side down, in the heated pan or griddle, as many as will fit in a single layer. Cook until browned, 4 to 5 minutes on each side. Repeat as needed until all the quesadillas are cooked.
Reheating: If not serving immediately, let cool completely and wrap each quesadilla tightly in aluminum foil. Refrigerate or freeze in resealable plastic bags. Reheat uncovered in a 325°F regular or toaster oven until warmed through, about 20 minutes if refrigerated, or about 30 minutes if frozen. Microwaving is not recommended, as the quesadillas will be soggy.