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 allow flavors to meld and juices to release.
02 - Place an egg roll wrapper on a clean surface with one corner pointing toward you. Spoon approximately 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 top corner. Brush the top edge with beaten egg and seal securely. Repeat with remaining wrappers and filling.
03 - Heat 2 inches of vegetable oil in a deep skillet or pot to 350°F. Working in batches, fry egg rolls for 2 to 3 minutes per side until golden brown and crispy. Transfer to a wire rack or paper towels to drain excess oil.
04 - Whisk together powdered sugar, milk, and vanilla extract in a small bowl until smooth and pourable. Drizzle generously over warm egg rolls immediately before serving.