01 - Combine diced peaches, granulated sugar, brown sugar, lemon juice, cinnamon, nutmeg, and cornstarch in a medium bowl. Mix thoroughly and allow the mixture to stand for 5 minutes to macerate, ensuring the flavors meld and the cornstarch begins thickening the fruit juices.
02 - Place an egg roll wrapper on a clean work surface with one corner pointing toward you. Spoon approximately 2 tablespoons of the peach filling near the bottom corner. Fold the bottom corner over the filling, then fold in the side corners. Roll the wrapper tightly away from you, forming a compact cylinder. Brush the final corner with beaten egg and press firmly to seal. Continue until all wrappers and filling are used.
03 - Pour vegetable oil to a depth of 2 inches in a deep skillet or heavy-bottomed pot. Heat the oil to 350°F. Working in batches to avoid overcrowding, carefully place the egg rolls in the hot oil. Fry for 2 to 3 minutes per side until golden brown and crispy throughout. Use a slotted spoon to transfer the finished rolls to paper towels, allowing excess oil to drain.
04 - Whisk together the powdered sugar, milk, and vanilla extract in a small bowl until completely smooth and free of lumps. The glaze should have a pourable consistency. Drizzle generously over the warm egg rolls immediately before serving.