Preheat a panini pan or a removable plate grill on medium heat. While the pan is preheating, begin to assemble the panini.
Lightly brush one side of each slice of bread with olive oil. Lay the bread flat on a plate, oiled side down.
Place the Swiss cheese on each slice of bread, covering the surface of the bread followed by a layer of prosciutto, being sure to cover the cheese. Place the arugula/spinach on top of the cheese on just one of the bread slices. Place tomato slices on top of arugula/spinach. Place the over-easy eggs on top of the tomatoes. Place the other slice of bread with cheese and prosciutto on top of the eggs - meat facing eggs.
Test the panini pan for the correct temperature by drizzling a few drops of water on the preheated surface. If the water "dances" on the surface the pan, it is ready.
Place the assembled panini in the pan and place the weight on top of the panini or lower the top grill plate.
Toast until the first side is golden brown and has grill marks over the surface of the bread. If using a grill pan, turn the panini over and toast on the other side. If using the removable plate grill, cook until panini has reached desired doneness on both sides.