In a small skillet heat the oil. Saute the onions in the oil until golden brown, about ten minutes, stirring. Season to taste with salt and pepper.
In a small bowl, combine the mayonnaise and mustard. Set aside.
Layer the sandwich as follows: rye bread, cheese, onion mixture, turkey slices, tomatoes, spinach. Spread desired amount of mayo across one side of the top bread slice. Place it mayo side down onto the spinach.
Grill the sandwich on a panini press. Serve hot.
Nutritional values are approximate and are based on 1/4 of the recipe for each sandwich. Refrigerate leftovers promptly and use within 4 days.