Preheat a skillet over medium heat and add the thick-cut bacon. Cook the bacon until crispy, about 6 to 8 minutes, then remove and let drain on paper towels.

In a mixing bowl, crack the eggs, add the milk, salt, and pepper. Whisk until combined and slightly frothy.

Drain most of the bacon grease from the skillet, leaving about 1 tablespoon. Pour the egg mixture into the skillet, cooking gently for about 3 to 5 minutes until just set.

Lay out two slices of bread, distribute the shredded cheddar cheese over each slice, then spoon the scrambled eggs over the cheese, followed by two strips of bacon.

Sprinkle remaining cheddar cheese on top of the bacon and place the other two slices of bread on top.

Spread the outside of each sandwich with unsalted butter.

Grill the sandwiches in a skillet over medium heat for about 3 to 4 minutes on each side, pressing down with a spatula.

Once golden brown, remove from heat and let cool slightly before slicing.

Garnish with fresh chives or green onions before serving.