In a medium bowl, whisk together the eggs, whole milk, salt, and black pepper until fully combined.
Heat a non-stick skillet over medium heat and add the butter. Once melted, pour in the egg mixture and cook, stirring gently, until the eggs are scrambled and cooked through, about 3-4 minutes. Remove from heat and set aside.
In a separate skillet, heat the vegetable oil over medium-high heat. Add the frozen hash browns and cook, stirring occasionally, until golden brown and crispy, about 6-8 minutes. Remove from heat.
Warm the flour tortillas in the microwave for about 15 seconds to make them pliable.
To assemble each crunchwrap, place one tortilla on a clean surface. In the center, layer 1/4 of the scrambled eggs, one cooked sausage patty, 1/4 cup of crispy hash browns, and 2 tablespoons of shredded cheddar cheese.
Fold the edges of the tortilla towards the center to create a sealed wrap, overlapping the folds as needed.
Heat a clean skillet over medium heat. Place the assembled crunchwrap seam-side down in the skillet and cook for 2-3 minutes until golden brown and crispy. Carefully flip and cook the other side for an additional 2-3 minutes.
Remove from the skillet and let cool for 1 minute before serving. Repeat with the remaining ingredients.