In a large skillet, melt the butter over medium heat. Add the brown sugar, cinnamon, and nutmeg. Stir continuously until the sugar has dissolved and the mixture becomes fragrant. Incorporate the diced apples and lemon juice. Cook, stirring occasionally, for approximately 5 minutes, or until the apples begin to soften but maintain their shape.

In a small bowl, whisk together the cornstarch and water until fully combined. Pour this slurry into the apple mixture along with the vanilla extract. Stir well and continue cooking for an additional 2 to 3 minutes, or until the mixture thickens to a consistency similar to traditional pie filling. Remove the skillet from heat and allow the mixture to cool slightly.

Lay one egg roll wrapper on a clean, dry surface in a diamond shape, with one corner pointing toward you. Place 1–2 tablespoons of the cooled apple filling just below the center. Lightly brush the edges with the beaten egg. Fold the bottom corner over the filling, then fold in both side corners. Roll tightly, sealing the top corner to secure. Repeat with remaining wrappers and filling.

In a deep skillet or saucepan, heat approximately 2 inches of vegetable oil to 350°F (175°C). Fry the egg rolls in small batches for 2 to 3 minutes per side, or until golden brown and crisp. Do not overcrowd the pan. Transfer to a paper towel–lined plate to drain excess oil.

In a shallow bowl, mix the granulated sugar and ground cinnamon. While the egg rolls are still warm, gently roll them in the cinnamon sugar mixture to coat evenly.

Serve warm with optional toppings such as caramel sauce, whipped cream, or a scoop of vanilla ice cream.