01 - Combine peaches, granulated sugar, brown sugar, lemon juice, cinnamon, nutmeg, and cornstarch in a medium bowl. Mix thoroughly and allow to stand for 5 minutes to macerate the fruit.
02 - Place an egg roll wrapper on a clean surface with one corner facing you. Spoon 2 tablespoons of peach filling near the bottom corner. Fold the bottom corner over the filling, fold in the sides, and roll tightly toward the opposite corner. Brush the top edge with beaten egg to seal securely. 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 crispy. Transfer to paper towels to drain excess oil.
04 - Whisk together powdered sugar, milk, and vanilla extract until smooth and free of lumps. Drizzle over warm egg rolls immediately before serving.