Place the potatoes in a cheesecloth or clean kitchen towel and squeeze to remove as much moisture as possible.
Heat a large non-stick skillet over medium heat. Once the skillet is hot, add the sausage and cook, breaking up the meat with a wooden spoon, until browned and cooked through, about 5 minutes. Transfer to a plate.
To the same skillet, add in the eggs and season with ½ teaspoon of the salt and the pepper. Using a rubber spatula stir the eggs slowly to scramble until the eggs are set but still wet, 1 to 2 minutes. Transfer to the plate with the sausage.
In a second large skillet, heat the olive oil over medium-high heat. Add the potatoes and spread out into an even layer, then seasoning with the remaining ½ teaspoon of salt. Cook undisturbed for 2 to 3 minutes, then flip and cook for another 2 to 3 minutes (it’s ok if the potatoes break up). Continue cooking, stirring occasionally, until the potatoes are browned and slightly crispy. Add the green onions along with the cooked sausage and scrambled eggs and stir to combine. Remove the skillet from heat.
Turn on the broiler to LOW. Add ½ cup sausage and ½ cup egg mixture to the center of each tortilla. Top each with ⅓ cup of the cheese, avocado slices, salsa, and cilantro. Working with one tortilla at a time, fold the ends over the filling and roll the tortilla until it overlaps over the filling to form a burrito. Add the burritos to a baking sheet, seam side down and broil until the tortillas are golden, about 5-6 minutes.