Preheat the oven to 325 degrees F (165 degrees C). Lightly grease a jumbo muffin tin.
Scramble egg whites, 4 eggs, 1/3 cup milk, and salt to taste in a bowl until light and fluffy. Fill each muffin tin compartment 1/2 full with egg mixture. Cover with aluminum foil.
Bake in the preheated oven until eggs are set, about 25 minutes.
Transfer eggs to a clean surface and immediately place 1 slice Cheddar cheese on each so they melt.
Mix baking powder, flour, and 2 teaspoons salt together in a large bowl.
Mix 4 eggs, syrup, sweetener, melted butter, and vanilla extract together in a bowl. Pour into the bowl with the baking powder mixture, adding 1 quart milk as you mix. Stir until pancake batter is the consistency of melted peanut butter.
Heat a griddle over medium heat and lightly coat with cooking spray. Pour approximately 1/4 cup batter onto the griddle, swirling the spoon to create a circular shape. Cook until bubbles form on the surface, 1 to 3 minutes. Flip only once and cook about 30 seconds more. Remove from griddle. Repeat with remaining batter.
Allow all ingredients to coo. Make sandwiches using 2 pancakes as the bun and an egg and cheese portion in the center. Wrap each sandwich individually in plastic wrap. Place sandwiches in a resealable plastic freezer bag and store in the freezer.