01 - Combine peaches, granulated sugar, brown sugar, lemon juice, cinnamon, nutmeg, and cornstarch in a medium bowl. Mix thoroughly and let stand for 5 minutes to macerate.
02 - Place an egg roll wrapper on a clean surface with a corner facing you. Spoon 2 tablespoons of peach filling near the bottom corner. Fold the bottom corner over the filling, tuck in both sides, and roll up tightly. Seal the final corner with beaten egg. Repeat with remaining wrappers and filling.
03 - Heat 2 inches of vegetable oil in a deep skillet or pot to 350°F. Fry egg rolls in batches for 2 to 3 minutes per side until golden brown and crisp. Remove with a slotted spoon and drain on paper towels.
04 - Whisk powdered sugar, milk, and vanilla extract until smooth and drizzle over warm egg rolls before serving.