Make the sauce: In a small bowl, stir together the mayonnaise, sour cream, diced pickled jalapeños, jalapeño juice, garlic powder, onion powder, chili powder, smoked paprika, cumin, and a pinch of salt. Mix until fully combined and smooth. Set aside.
Cook the hash browns: Follow the package directions — bake on a sheet pan at 425°F for about 12 minutes, flipping halfway, until deeply crispy on both sides. You can also pan-fry in a dry nonstick skillet over medium heat. Set aside on a paper-towel-lined plate.
Cook the sausage: In a small skillet over medium heat, cook the bulk sausage, breaking it up with a spatula, until no longer pink and browned edges form — about 5 minutes. Drain off excess grease and transfer to a small bowl.
Scramble the eggs: Wipe out the same skillet and add butter over medium-low heat. Pour in beaten eggs, season with salt, and cook, stirring gently with a rubber spatula. Pull off heat when still slightly soft and custardy.
Assemble the wraps: Lay one tortilla flat. Spread about 1/4 of the sauce in a generous circle in the center, leaving a 1-inch border. Place one hash brown patty in the center. Top with 1/4 of the sausage, 1/4 of the eggs, and 1/4 of the shredded cheddar.
Fold them tight: Fold the tortilla side closest to you into the center, then fold in the right side, holding the first fold. Continue folding in small sections until the filling is completely enclosed. Press gently to seal. Place seam-side down.
Sear the wraps: Heat a large nonstick skillet over medium-low heat. Place two wraps seam-side down in the pan. Cook for 3-4 minutes until deeply golden and crisp. Flip carefully and cook the other side for 2-3 minutes until golden and cheese has melted.
Rest and serve: Transfer seared wraps to a cutting board and let rest for a minute before slicing. Slice in half or quarters and serve immediately. Repeat with remaining wraps.