Melt butter in a medium skillet over medium heat. Add diced apples, brown sugar, granulated sugar, cinnamon, nutmeg, and lemon juice.
Cook 6–8 minutes until the apples soften but hold their shape. Stir in flour to thicken, cook 1–2 minutes more, then remove from heat. Mix in vanilla and let cool slightly.
Place an egg roll wrapper on a clean surface in a diamond shape. Spoon about 2 tablespoons of filling into the center.
Fold the bottom corner over the filling, tuck in the sides, brush the top corner with beaten egg, and roll up snugly to seal. Repeat with the remaining wrappers.
Heat 2 inches of vegetable oil in a skillet to 350°F (175°C). Fry egg rolls in batches for 2–3 minutes per side until golden and crisp, then drain on paper towels. For a lighter option, brush with oil and air fry at 375°F for 10–12 minutes until golden.
Dust the warm egg rolls with powdered sugar and serve right away with caramel sauce for dipping or a scoop of vanilla ice cream.