Add 1 tablespoon vegetable oil to a large skillet and place over medium heat.
Add potatoes, ½ teaspoon salt, and ¼ teaspoon black pepper and cook, tossing occasionally, until golden brown, about 10 minutes. Transfer to a plate and set aside.
To the same skillet, add the remaining 1 tablespoon vegetable oil and chopped onion. Cook for 2-3 minutes, until slightly golden.
Add sausage and cook for 8 more minutes, stirring often, until browned. Transfer mixture to a plate and set aside.
Add eggs, ¼ teaspoon salt, and ¼ teaspoon black pepper to the skillet.
Cook, scrambling with a rubber spatula, for 2 minutes, until set. Transfer to a plate and set aside.
Add each tortilla to the skillet one at a time and warm for 10-15 seconds on each side.
Divide cooked potatoes, sausage and onion, and eggs between the center of each tortilla.
Top with cheddar cheese, chopped green onion, and avocado slices.
Wrap burritos tightly.
Place burritos seam side-down in a hot skillet until slightly brown, about 15 seconds. Flip and brown on the other side.