Preheat a medium skillet over medium heat. Once hot, add thick-cut bacon slices and cook for about 5-7 minutes, flipping occasionally, until crispy and golden brown. Use tongs to transfer the bacon to a plate lined with paper towels, allowing the excess grease to drain, while keeping the flavorful bacon fat in the skillet for later use.

In a medium bowl, whisk together the large eggs, a splash of milk, and a pinch of salt and pepper until well combined. The goal is to achieve a light and frothy mixture. Set the bowl aside while you prepare the skillet, ensuring the eggs will be fluffy when they hit the heat.

With the skillet still over medium heat, leave about 1 tablespoon of the bacon grease. Pour in the egg mixture and let it sit undisturbed for about 30 seconds. Then, gently stir with a spatula for 3-5 minutes, until the eggs are softly set and slightly moist.

On a clean cutting board, place two slices of sourdough bread side by side. Layer shredded cheddar cheese generously on each slice. Next, evenly distribute the fluffy scrambled eggs and crispy bacon over the cheese. Top with a bit more cheese before capping each sandwich with the remaining bread slices.

Spread a thin layer of unsalted butter on the outer sides of both sandwich halves to ensure an even beautiful golden crust.

In the same skillet, increase the heat to medium-high. Carefully place the sandwiches into the skillet and grill for about 3-4 minutes on each side or until the bread is golden brown and the cheese has melted beautifully.

Once grilled to perfection, remove the sandwiches from the skillet and let them rest on a cutting board for about 5 minutes.

Plate the sandwiches with your choice of sides, perhaps some fresh fruit or crispy hash browns, and serve.