Prepare the hash brown patties according to the package directions until golden and crisp. An air fryer works especially well, but you can also bake or pan-fry them. Keep them warm while preparing the remaining fillings.
Heat a large skillet over medium heat. Add the chorizo and diced onion and cook for 5–7 minutes, breaking the sausage apart with a spatula, until the chorizo is fully cooked and the onion is tender.
Increase the heat to medium-high and continue cooking for another 1–2 minutes, stirring occasionally, until the chorizo develops lightly browned and crispy edges. Transfer the mixture to a plate, leaving some of the flavorful chorizo drippings in the skillet.
Crack the eggs into a bowl and whisk with the salt and black pepper. Reduce the skillet heat to medium-low and pour in the eggs.
Cook the eggs slowly, stirring gently with a spatula, until they are softly set and fluffy, about 3 minutes. Remove them from the heat while they are still slightly moist, as they will continue cooking from the residual heat.
Warm the flour tortillas in a dry skillet or microwave until soft and flexible. If using uncooked tortillas, cook them according to the package directions before assembling the burritos.
Place one crispy hash brown patty in the center of each tortilla. Divide the scrambled eggs evenly between the tortillas, then add the shredded cheese.
Top each burrito with the cooked chorizo and onion mixture. Keep the filling in a compact strip through the center so the tortillas are easier to roll.
Fold the bottom edge of each tortilla over the filling, fold the sides inward, then roll tightly toward the top to form a burrito.
For a crisp, sealed exterior, place the burritos seam-side-down in a preheated sandwich press or skillet. Cook for about 2 minutes, turning if necessary, until the tortilla is lightly golden and the cheese has melted.
Serve the burritos hot with salsa, avocado, fresh cilantro, hot sauce, or your favorite breakfast toppings.