01 - Combine diced peaches, granulated sugar, brown sugar, lemon juice, ground cinnamon, nutmeg, and cornstarch in a medium bowl. Stir thoroughly to coat peaches evenly. Let mixture stand for 5 minutes to allow flavors to meld and cornstarch to activate.
02 - Position an egg roll wrapper on a clean work surface with one corner pointing toward you. Place approximately 2 tablespoons of peach filling just above the bottom corner. Fold the bottom corner over the filling, fold in the side corners, then roll tightly upward. Seal the final flap with beaten egg. Continue until all wrappers are filled.
03 - Pour vegetable oil to a depth of 2 inches in a deep skillet or Dutch oven. Heat oil to 350°F. Fry egg rolls in batches of 4-5 for 2-3 minutes per side until golden brown and crispy. Transfer to paper towel-lined plate to drain excess oil.
04 - Whisk together powdered sugar, milk, and vanilla extract in a small bowl until smooth and creamy. Drizzle generously over warm egg rolls immediately before serving.