In a medium bowl, whisk together soy sauce, vinegar, Sriracha, sugar, ginger, and sesame oil; set dipping sauce aside.
Make pot stickers: In a food processor, process scallions, carrot, egg white, and half the shrimp until a paste forms. Transfer to a medium bowl. Roughly chop remaining shrimp and add to bowl. Stir in 3 tablespoons dipping sauce.
Working with one wonton wrapper at a time, place 1 scant tablespoon shrimp mixture in center. With a wet finger, moisten edge of wrapper, then fold over filling and press to seal. Cover filled dumplings with a kitchen towel while you work.
Lightly coat a large nonstick skillet with cooking spray and heat over high. In batches, cook dumplings until golden brown on both sides, 1 to 2 minutes per side. Carefully add 1/2 cup water to skillet, cover, and cook until water has almost evaporated and dumplings are tender, 3 minutes. Uncover and cook until remaining water evaporates (reduce heat to medium if dumplings are overbrowning). Transfer pot stickers to a plate. Wipe skillet clean between batches. Add scallion greens to dipping sauce if desired and serve alongside pot stickers.
Freeze uncooked dumplings in a single layer, then store in freezer bags, up to 3 months.