To assemble sandwiches, spread four pieces of bread with one tablespoon (15 milliliters) of the Dijon, then top each with two slices of ham, sprinkle over 1/4 of the Gruyère, and top with another piece of bread.
In a shallow dish, whisk together eggs and milk, then season with salt and pepper.
Heat a large skillet over medium heat, then add one to two tablespoons (15-30 milliliters) of butter. Once butter is melted and skillet is hot, dip sandwiches in egg mixture to coat, letting any excess dip off. Place in skillet and cook, working in batches of one or two at a time, adding butter as needed. Cook sandwiches, flipping once, until cheese is melted and bread is golden brown on both sides.
Slice sandwiches in half and serve with a small side of red currant jelly for dipping.