Heat a large griddle or cast iron skillet over medium heat.
Prepare the sandwiches by spreading four slices of bread with the preserves. Top each slice of bread with a slice of cheese, two slices of folded prosciutto, then another slice of cheese. Spread the remaining slices of bread with Dijon and set in place.
Beat the eggs with the milk or cream, Parmigiano Reggiano and parsley. Nest the butter in a paper towel and wipe it across the hot pan. Dip the sandwiches into the egg mixture and griddle the sandwiches until deep golden, 3-4 minutes on each side.
Toss the arugula with the lemon juice, EVOO (eyeball it), salt and pepper.
Cut the sandwiches from corner to corner and serve alongside the salad.