Preheat the oven to 400°F. Arrange the tortillas on a baking sheet and set aside. Chop the onion.
In a large nonstick skillet over medium-high heat, heat 2 tablespoons of the oil. Add the onion, lower the heat to medium-low, and cook until translucent, 5 to 7 minutes.
Add the frozen hash browns, season with salt and black pepper to taste, and increase the heat to medium-high. Cook until the hash browns turn brown and crispy, 3 to 5 minutes on each side, adding more oil, 1 teaspoon at a time, if necessary. Transfer the hash brown mixture to a paper towel-lined plate.
Wipe the skillet with a paper towel, add 1 teaspoon of the oil, and cook the bacon strips until crispy, 3 to 4 minutes. Transfer to a second paper towel-lined plate. Leave the strips whole or crumble if desired.
Divide the hash brown mixture and bacon equally among the tortillas. Top each with ¼ cup of the cheese. Place the baking sheet in the oven while you prepare the eggs.
Place the eggs in a large bowl and whisk for 30 seconds. Add salt and black pepper to taste.
With a paper towel, wipe the skillet again and place it over medium-low heat. Add the butter and, as it melts, swirl to coat.
Cook the whisked eggs in the skillet for 30 seconds to 1 minute without disturbing.
Between 1 and 2 minutes, when the eggs are about three-quarters of the way done, use a rubber spatula or wooden spoon to fold the eggs and form big, fluffy curds. Turn off heat but keep the eggs in the skillet until just set and still moist.
Quarter the eggs and place each serving next to the hash browns on the tortillas. To each tortilla, add 2 tablespoons of green chiles and 1 tablespoon of red chiles.
Fold the left and right sides of each tortilla over the filling, then roll tightly, from the bottom up, over the filling. Repeat with the remaining burritos. Top with the two sauces to taste.