Let the 4 hard-boiled eggs cool enough to handle, peel them, and cut into roughly 1/2-inch pieces; place the chopped eggs in a medium bowl.
Measure 3/4 cup finely grated cheese and set aside. Put 1/4 cup of that cheese into a small bowl. In the small bowl whisk together 2 tsp Dijon mustard and 3 T mayo, then stir in 2 T finely chopped green onion, the 1/4 cup grated cheese, and 1/2 tsp Spike Seasoning.
Fold the mayo mixture into the chopped eggs. Taste and add salt and freshly ground black pepper to your liking. This is the egg salad filling.
Spread the 1 T butter evenly on one side of each of the 4 tortillas (the buttered side will be the outside of the quesadillas). Reserve the remaining 1/2 cup grated cheese for assembling.
Heat a nonstick skillet over medium heat. Place one tortilla in the skillet buttered-side down. Sprinkle 2 tablespoons of the reserved grated cheese over the tortilla.
Spoon about half of the egg salad (roughly half the mixture) onto the cheese, leaving a 1/2-inch border around the edge so the filling won’t spill when cooking. Sprinkle another 2 tablespoons of grated cheese over the egg salad. Top with a second tortilla, buttered-side up.
Cook the quesadilla 3–5 minutes per side, or until the tortilla is golden brown and the cheese is melted. Lift an edge to check browning before flipping. Flip carefully with a spatula and cook the second side.
Transfer the cooked quesadilla to a cutting board, cut in half or into wedges, and keep warm. Repeat steps 5–7 with the remaining two tortillas and the remaining egg salad to make a second quesadilla.
Serve hot.