Stir together the mayonnaise, Dijon, cornichons, garlic, salt, and pepper in a small bowl.
Spread the mayonnaise across the tortilla.
Cover half the tortilla with ½ cup shredded cheese.
Layer ham on top of the cheese.
Fold the tortilla in half.
Melt the butter in a large cast iron skillet over medium-low heat.
Once the butter is melted, arrange the remaining ½ cup of shredded cheese across the skillet in the shape of the folded tortilla.
Add the quesadilla on top of the cheese and press down firmly. Press and cook until the cheese is melted into the bottom of the quesadilla, about 3 minutes.
Flip and cook the other side until the insides are completely melted, about 2 minutes longer.
Remove from the skillet, cut into triangles, and serve.