Mix the chopped ham, turkey, spinach, artichoke hearts and Gruyère in a bowl. Spoon mixture onto 4 slices of bread and spread it out evenly with a spatula. Place remaining slices of bread over the top to complete the sammies.
Melt butter in a griddle or cast-iron pan over medium-high heat. Dip each sammy into the beaten egg mixture and place in the pan. Cook until golden and cheese is melted.