In a medium bowl, crack the eggs and season them with salt and pepper. Whisk until well combined.
Heat a non-stick skillet over medium heat and add 1 tablespoon of butter. Once the butter is melted, pour in the whisked eggs and cook, stirring occasionally, until they are softly scrambled. Remove from heat and set aside.
Lay the flour tortillas on a clean work surface. Spoon a portion of the scrambled eggs onto the center of each tortilla. Top with an equal amount of shredded cheddar cheese and any optional fillings you prefer.
Carefully fold the tortillas in half, pressing gently to seal in the filling.
In the same non-stick skillet, heat the remaining tablespoon of butter or a small amount of oil over medium heat. Place 2 to 3 filled tortillas in the skillet, making sure not to overcrowd the pan.
Cook the taquitos for 2-3 minutes on each side, or until the tortillas are golden brown and crispy, and the cheese has melted. Use a spatula to carefully flip the taquitos halfway through cooking.
Remove the cooked taquitos from the skillet and place them on a paper towel-lined plate to drain any excess grease. Repeat the process with the remaining tortillas, adding more butter or oil to the skillet as needed.
Serve the Egg and Cheese Taquitos warm with your favorite dipping sauces, such as salsa, sour cream, or guacamole. Enjoy!