Dice the potatoes into small cubes, season with salt and pepper, and fry in a hot pan with about 2 tablespoons of oil over medium heat. Cook for 10 minutes, stirring occasionally, until the potatoes are golden brown and crispy on the outside, and soft inside. (You can check doneness by piercing a potato cube with a fork—it should go in easily.)
While the potatoes are cooking, crack the eggs into a bowl, add a pinch of salt and pepper, and whisk until well combined. Heat a separate pan over medium-low heat, add a little butter or oil, and pour in the eggs. Stir gently and frequently to cook the eggs until soft and fluffy, being careful not to overcook them.
Lay a tortilla flat on a clean surface. Add about 1/4 cup of the cooked potatoes and 1/4 cup of scrambled eggs to the center. Sprinkle with 2 tablespoons of cheese (cheddar or your preferred cheese) for extra flavor.
Fold in the sides of the tortilla, then roll tightly from the bottom, making sure the sides are tucked in to keep the filling secure. The key is to roll it firmly but not too tight to avoid squeezing the fillings out.
Heat a skillet over medium heat. Place the burrito seam-side down in the pan and cook for 2-3 minutes on each side, until the tortilla is crispy and golden brown, and the cheese is melted inside. Flip carefully with a spatula to avoid spilling the filling.
Remove the burrito from the skillet, slice in half if desired, and serve warm with your favorite condiments like salsa, guacamole, or sour cream.