In large bowl, beat eggs, milk, salt, and pepper; keep refrigerated until service.
For each sandwich, pour 1/2 cup egg mixture in buttered 6-inch pan over medium heat. Cook until firm throughout with no visible liquid remaining.
Layer each sandwich: bread, cheddar slice, 2 sausage patties, eggs, cheddar slice, bread.
In spray-coated Panini machine, cook sandwiches until cheese melts and sandwiches are deep golden brown. (If using a grill pan, press/weigh-down sandwiches using smaller heavy pan.) Serve immediately.
Note: Do not let uncooked egg mixture remain at room temperature for longer than one hour (including preparation and service time).
*If using frozen or liquid whole egg product.