9 large sheets spring roll pastry wrappers, cut into quarters (See Cook’s Note)
1 large egg, beaten
Peanut or canola oil, for frying
For the sauce: Whisk together 1/3 cup water, the sugar, ketchup, vinegar, Sriracha and cornstarch in a small saucepan over medium-high heat, and bring to a boil. Remove from the heat when the sauce begins to thicken, 30 to 45 seconds. Let cool.
For the spring rolls: Meanwhile, heat a large skillet or wok over medium-high heat.
Add the oil, ginger and garlic, and cook, stirring constantly, until aromatic, about 30 seconds. Add the cabbage and carrots, and cook, stirring, until wilted, about 1 minute.
Add the shrimp, scallions, mirin, soy sauce, 1/4 teaspoons salt and a few cracks of black pepper, and cook, stirring, until the shrimp turn pink, about 2 minutes. Remove from the heat, and stir in the sesame oil. Let the mixture cool for 15 minutes. Fold in the bean sprouts. When completely cool, roughly chop the mixture and set aside.
Put 1 spring roll wrapper on a work surface with a point facing you (it should be a diamond shape). Put about 1 tablespoon of the shrimp mixture in the center of the wrapper, and spread to make a 1-inch log.
Fold the bottom of the wrapper over the filling, rolling tightly to keep the spring rolls firm. Fold the sides of the wrapper over the filling.
Brush the top edge of the wrapper with egg.
Roll up tightly, pressing to seal the edge. Put finished rolls, seam-side down, on a plate or baking sheet until ready to fry. Repeat with the remaining wrappers and filling.
Fill a large heavy-bottomed pot or Dutch oven with 3 inches of oil, and attach a deep-fry thermometer to the side. Heat the pot over medium-high heat to 375 degrees F. Fry the spring rolls in batches, 4 or 5 at a time, until lightly golden, about 2 minutes. Remove with a slotted spoon to a paper towel-lined plate to absorb excess oil. Serve with the dipping sauce.