Place the center-cut bacon in a cold non-stick skillet and turn the heat to medium, cooking until the fat has rendered and the bacon is crisp.
Remove the bacon and set aside on a paper towel, then lightly toast the sourdough slices in the same pan until golden brown.
In a small bowl, whisk together the whole egg, liquid egg whites, sea salt, and black pepper until well combined.
Wipe the skillet clean and melt the ghee over medium-low heat.
Pour the egg mixture into the skillet and cook slowly, using a spatula to gently push the cooked edges toward the center to create soft, fluffy curds.
When the eggs are nearly set, sprinkle the sharp cheddar cheese over the top and cover the pan for 30 seconds to melt.
Layer the cheesy eggs onto one slice of toasted sourdough, followed by the crispy bacon and fresh baby arugula.
Top with the remaining slice of bread, slice in half, and serve immediately.