In a small bowl, stir together the spread ingredients: Greek yogurt, Dijon, and salt. Spread a thin layer over one side of each of the two pieces of bread.
Place the bread on a plate, and butter the opposite side of each slice, making sure the butter reaches all the way to the edges.
Assemble the sandwiches: Place the cheeses in an even layer on top of the Dijon side of the bread slice (it will look like a lot of cheese, but go with it). Place the second slice of the bread on top of the cheese, Dijon-side down. If you are using cheese slices (vs. shredded cheese) and any of the cheese pokes out around the outsides of the bread, trim the extra cheese away (to avoid waste, stack the trimmings in the center of the sandwich).
Secure the sandwich by sticking a toothpick into two of the opposite corners of the bread.
Slide out the basket and place it on a steady, heat-proof surface. Carefully lower the sandwich into it, then slide the basket back into the air fryer.
Place in the air fryer and cook on the first side for 4 to 6 minutes, until toasted and deep golden on top (the amount of time you need will vary based on your air fryer model).
Slide out the basket and place it on a steady surface once more. Remove the toothpicks and with a spatula or tongs, carefully flip the sandwich over. Return the basket to the air fryer and continue cooking for 3 to 4 additional minutes, until the bread is toasted and the cheese is hot and melted.
Transfer to a plate or cutting board. Let cool a few minutes, then slice and enjoy.
TO REHEAT: Grilled cheese is best enjoyed right away, but it can be re-toasted in the air fryer at 350 degrees F.