Heat a large griddle or cast-iron skillet over medium heat.
Prepare the sandwiches by spreading 4 slices of the bread with preserves. Top with a slice of cheese, 2 slices of folded prosciutto, then another slice of cheese. Spread the remaining bread slices with Dijon and set in place over the cheese.
In a large shallow bowl, beat the eggs with the milk or cream, Parmesan, and parsley. Nest the butter in a paper towel and wipe it across the hot pan. Dip the sandwiches into the egg mixture and cook the sandwiches until deep golden, about 3 to 4 minutes on each side.
Toss the arugula with the lemon juice, extra-virgin olive oil, and salt and pepper, to taste, in a large serving bowl. Cut the sandwiches from corner to corner and serve alongside the salad.