Whisk together the Dijonnaise and cranberry relish in a small bowl and season with salt and pepper.
Put the bread on a flat surface and spread each slice with some of the Dijonnaise mixture. Top 2 of the slices of bread with 1 slice of the cheese, ham, turkey another slice of cheese and pickles (in that order). Cover with the remaining bread, Dijonnaise side down.
Heat a hot skillet over medium heat or a panini press. Wrap a brick in heavy duty aluminum foil (if not using a panini press).
Spread 1 tablespoon of the mayonnaise on the top of each sandwich and add to the skillet, if using, or in a panini press, mayonnaise side down. Put the brick on top of the sandwiches and cook until the bottom is golden brown, about 2 minutes. Remove the brick, spread the top of the sandwiches with the remaining mayonnaise, turn over, and return the brick to the sandwiches. Cook until the bottom is golden brown and the cheese has melted, about 2 minutes longer. Cut in half, transfer to serving plates and serve warm.