In a small bowl, mix together the jam, mustard and rosemary. Lay out the slices of bread. Divide the mustard mixture evenly over the 8 slices of bread and spread almost to the edges. Put a slice of cheese on each piece of bread, followed by 2 slices of prosciutto. Put together 2 of the bread slices for each sandwich.
In a medium bowl, whisk together the eggs. Add the heavy cream and salt and whisk to combine.
Heat a large skillet over medium heat. Add the oil to the pan and heat for 1 minute. Working with one sandwich at a time, dip both sides of the sandwich in the batter and place directly into the hot pan. Repeat with the remaining sandwiches. Cover with a lid and allow to cook until the first side is golden brown, 3 to 4 minutes. Remove the lid, flip each sandwich and continue to cook on the remaining side until golden brown and the cheese is melted, approximately another 4 minutes. (Or cook in batches if needed.)
Sprinkle one side of each sandwich with powdered sugar and serve warm.