Pre-heat oven to 200° F.
For the butter mixture; in a mixing bowl add the room temperature butter and mayonnaise and whisk together until well combined. Reserve and set aside.
Sandwich assembly: Line large baking sheet trays with parchment paper.
Brush each slice of bread with a thin even layer of butter mixture and place in columns of two on parchment paper lined baking sheet tray buttered side down.
Spread cheese evenly over each slice of bread.
On one column top with sliced ham (the sandwich bottom) and the other column, the sliced apples (the sandwich top) in a thin layer. You may have to do in small batches if you do not have a large griddle.
On the griddle or large sauté pan over low to medium heat, add both the top and bottom pieces of bread buttered side down leaving both open faced, do not put together to form a sandwich yet.
Check bottom of slice of bread with spatula to check desired browning.
Keep heat low to medium because if it is too high the bread will brown or burn before the cheese melts.
It may take a few practices to get the correct browning temperature.
When the bread is golden brown flip the top slices with apple and top the bottom pieces of bread with the ham forming a sandwich.
Give each sandwich a gentle press, and cook to ensure golden
browning is achieved. Flip it over and cook for a few minutes more.
Place the finished sandwiches on the baking sheet in warming oven.
Slice and serve immediately.