Pour hot water over bean sprouts in a small bowl; let stand 5 minutes. Drain.
Meanwhile, in a Dutch oven, cook chicken over medium heat until no longer pink, 6-8 minutes, breaking into crumbles. Add green onions, ginger and garlic. Cook 1 minute longer; drain. Stir in 1/2 cup Chinese-style sauce, fish sauce and soy sauce; transfer to a large bowl. Wipe pan clean.
In the same pan, cook and stir coleslaw mix, spinach and drained bean sprouts until crisp-tender, 4-5 minutes. Stir into chicken mixture. Cool slightly.
Preheat air fryer to 400°. With one corner of an egg roll wrapper facing you, place 1/3 cup filling just below center of wrapper. (Cover remaining wrappers with a damp paper towel until ready to use.) Fold bottom corner over filling; moisten remaining wrapper edges with water. Fold side corners toward center over filling. Roll egg roll up tightly, pressing at tip to seal. Repeat.
In batches, arrange egg rolls in a single layer in greased air-fryer basket; spritz with cooking spray. Cook until golden brown, 8-12 minutes. Turn; spritz with additional cooking spray. Cook golden brown, 4-6 minutes longer. Serve with remaining Chinese-style sauce.