With a rasp grater, finely grate Parmesan onto a plate (you should have 1 cup). Brush one side of bread slices with oil and press into Parmesan, coating completely. Place bread, cheese side down, on a work surface. Spread half the slices with mustard and the other half with jam. Assemble sandwiches with bread, fontina, and arugula. In batches, in a nonstick skillet, cook sandwiches over medium until bread is crisp and golden and fontina is melted, 2 to 3 minutes per side.
Cheese both inside and out turns this diner standby on its head.