01 - Combine diced 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 the mixture to thicken slightly.
02 - Place an egg roll wrapper on a clean work surface with one corner pointing toward you. Spoon approximately 2 tablespoons of peach filling onto the lower third of the wrapper. Fold the bottom corner over the filling, fold in the side corners, and roll tightly toward the top corner. Brush the final corner with beaten egg and press firmly to seal. Repeat with remaining wrappers and filling.
03 - Pour vegetable oil to a depth of 2 inches in a deep skillet or heavy pot. Heat oil to 350°F. Fry egg rolls in batches of 4-5 for 2-3 minutes per side until golden brown and crispy. Use a slotted spoon to transfer to paper towels to drain excess oil.
04 - Whisk together powdered sugar, milk, and vanilla extract in a small bowl until smooth and free of lumps. The glaze should have a thick, pourable consistency. Add additional milk 1 teaspoon at a time if too thick, or more powdered sugar if too thin.
05 - Drizzle warm egg rolls with vanilla glaze immediately before serving. For best texture and flavor, serve while still slightly warm. Alternatively, allow guests to add glaze according to preference.