In a small bowl, stir together the mayonnaise and Dijon mustard until smooth; set aside.
Warm the stuffing and gravy separately in a microwave or small saucepan until hot but not boiling.
Spread softened butter on the outside of each bread slice. Spread the mayo-Dijon mixture on the inside of each slice.
For each sandwich, layer turkey, warm stuffing, and a slice of Swiss on one bread slice. Spread cranberry sauce on the other slice, then add arugula.
Close the sandwiches (buttered sides facing out). Heat a large skillet over medium heat.
Cook the sandwiches for 3 to 4 minutes per side, pressing gently, until the bread is golden and the cheese is melted.
Slice and serve with warm gravy for dipping.
Notes
Variations: Add crispy bacon and avocado for a club-style twist; swap Swiss for sharp cheddar; toast on a panini press for extra crunch; or add pickled red onions for tang. Use sturdy bread to hold juicy fillings.