In a large skillet, melt the butter over medium heat. Add the diced apples, granulated sugar, cinnamon, nutmeg, and lemon juice.
Cook, stirring occasionally, until the apples are tender and the mixture has thickened, about 8-10 minutes. Let the apple filling cool completely.
Lay an egg roll wrapper on a clean surface with one corner pointing towards you. Spoon about 2 tablespoons of the cooled apple filling onto the center of the wrapper.
Fold the bottom corner over the filling. Fold in the left and right corners. Roll up tightly to form an egg roll shape.
Dip your finger in the beaten egg and moisten the top corner of the wrapper. This will help seal the egg roll. Press to seal.
Repeat with the remaining wrappers and filling.
In a deep skillet or Dutch oven, heat the vegetable oil over medium-high heat until it reaches about 350°F (175°C).
Carefully place 2-3 egg rolls in the hot oil, being careful not to overcrowd the pan. Fry for 2-3 minutes per side, or until golden brown and crispy.
Remove the egg rolls from the oil with a slotted spoon and place them on a wire rack to drain. Repeat with the remaining egg rolls.
Serve warm, with caramel sauce or whipped cream for dipping, if desired.
Notes
For an extra sweet treat, you can sprinkle the fried egg rolls with cinnamon sugar while they are still warm.