1Heat 1 tablespoon butter in a medium nonstick skillet over medium heat until melted and foaming. Add diced onion and bell pepper, cooking until softened and translucent, about 4-5 minutes. Remove vegetables to a plate and set aside.
2Cook breakfast sausage in the same skillet over medium-high heat, breaking into small crumbles with a wooden spoon until browned and fully cooked, about 6-7 minutes. Transfer to the plate with vegetables.
3Add thawed hash browns to the skillet and spread in an even layer. Cook without stirring until golden brown on the bottom, about 3-4 minutes, then flip and cook until crispy, another 2-3 minutes. Season with a pinch of salt and transfer to the plate.
4Crack eggs into a bowl and whisk thoroughly with salt and pepper. Wipe skillet clean and melt remaining 1 tablespoon butter over medium-low heat. Pour in eggs and gently scramble, stirring slowly with a spatula until soft curds form but eggs remain slightly wet, about 2-3 minutes. Remove from heat immediately.
5Warm tortillas in a dry skillet for 15-20 seconds per side until pliable and lightly toasted, or microwave wrapped in a damp paper towel for 20 seconds.
6Lay one warmed tortilla flat on a clean work surface. Sprinkle ⅓ of the cheese in the center, creating a rectangular base about 4 inches wide. Layer half the scrambled eggs over the cheese, followed by half the hash browns, sausage, and vegetable mixture. Top with 1½ tablespoons salsa and sprinkle with another portion of cheese.
7Fold the bottom edge of the tortilla up over the filling, then fold in both sides tightly. Roll upward to create a compact wrap, keeping tension to prevent gaps. Repeat with second tortilla and remaining ingredients.
8Heat vegetable oil in the skillet over medium heat. Place wraps seam-side down and cook until golden brown and crispy, about 2 minutes. Flip and cook the opposite side until equally crispy, another 1-2 minutes.
9Slice each wrap diagonally on a bias and serve immediately with remaining salsa on the side for dipping.