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Heat oil in a medium skillet over medium-high heat. Add pork and garlic; cook until well browned, about 3 minutes, stirring occasionally. Add cabbage and scallions; cook, stirring occasionally, until the cabbage is starting to wilt, about 3 minutes. Stir in lime juice and soy sauce. Remove from heat.
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Lay out 3 egg roll wrappers with 1 point away from you; brush 1 side completely with egg. Place 3-4 tablespoons of the filling in the center of each wrapper. Roll the point closest to you over the filling; then bring the side points into the middle. Continue rolling away from you to form a tight seal around the filling. Repeat the process with the remaining wraps and filling.
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Place the egg rolls in the basket of an air fryer and coat with cooking spray. Cook at 390°F until golden brown, about 10 minutes, turning halfway through.