Preheat oven to 350 degrees F. Heat a large griddle pan over medium heat with vegetable oil.
In a medium bowl, whisk the eggs, cream, sugar, salt, and cayenne until blended. Pour into an 8 by 8-inch baking pan.
Spread some of the barbecue sauce onto each slice of bread. On 4 pieces bread, add 1 slice each of cheese, ham, and turkey, and repeat layer. Top with the other pieces of bread and press the sandwiches together with your hands. Dip into the wet batter turn to coat.
Add the battered sandwiches to the hot griddle and cook until golden brown on 1 side. Flip and continue cooking the other side until golden. Remove to a rack-lined sheet tray. Put into the preheated oven and bake until the cheese melts, about 5 more minutes. Remove to a cutting board and slice into triangles. Dust with Neely's dry rub, if desired, and serve.