Cut sausage links in half lengthwise, then cut each half into 1/2-inch slices.
Spray a nonstick skillet with cooking spray and heat over medium heat. Add sliced sausage and cook, stirring occasionally, until beginning to brown, around 2 to 3 minutes.
Set aside 1 tablespoon of chopped green onions (for garnish); add remainder to the sausage. Cook, stirring occasionally, until onions begin to soften and sausage is fully browned, another 2 to 3 minutes. Watch carefully so the onions don't burn.
Reduce the heat to medium-low, add the egg whites, and cook, stirring occasionally, just until set. Remove from heat and set aside.
While the eggs are cooking, place the green chile sauce in a small saucepan. Stir in lime juice and bring to a low boil, stirring occasionally.
Wrap tortillas loosely in a damp paper towel and microwave for 15 to 20 seconds, just until warm.
Place one tortilla on a clean work surface and add 1/2 of the filling in a tight log shape, about 1 1/2 inches from the bottom edge and sides.
Fold the bottom edge up over the filling. Bring both sides in snugly.
Roll tightly from the bottom to form a secure burrito.
Repeat with remaining tortilla and filling.
Place each rolled burrito seam-side down on a dinner plate. Top with a generous amount of sauce; sprinkle with grated cheese and freshly ground pepper. Garnish with crema or sour cream (if using) and chopped green onion. Serve immediately while warm and melty.