In a large non-stick skillet, cook the ground beef over medium heat until no longer pink, approximately 8-10 minutes. Transfer the cooked beef to a medium bowl lined with paper towels and set aside. Wipe the skillet clean with a paper towel.

In the same skillet, melt 2 tablespoons of butter over medium-high heat. Pour the beaten eggs into the pan. Stir occasionally to make scrambled eggs until the liquid is almost gone, about 4-5 minutes. Remove the pan from the heat.

Remove the paper towel from under the beef and combine the scrambled eggs with the beef in the bowl. Gently mix them together, ensuring an even distribution. Wipe the skillet clean and heat it over medium heat.

Butter one side of each slice of sourdough bread. Place a slice of bread in the skillet, buttered side down.

Top the bread with a slice of Colby-Jack cheese. Spoon about 1/2 cup of the beef-egg mixture over the cheese. Add another slice of cheese on top of the mixture, then place another slice of buttered bread on top, butter side facing up.

Cook until the bread turns golden brown, about 3-4 minutes. Carefully flip the sandwich and cook until the other side is golden brown, approximately 3-4 minutes more.

Transfer the sandwich to a cutting board and cut it in half if desired. Repeat the process with the remaining ingredients.

Serve the breakfast grilled cheese sandwiches immediately. Enjoy!