Melt the butter in a skillet over medium heat.
Add apples, both sugars, 1.25 tsp cinnamon, nutmeg, and salt. Cook until tender, 5–7 minutes.
Stir in lemon juice. Mix cornstarch with 1 tbsp water, then stir into apples. Simmer to thicken. Cool.
Combine 0.33 cup sugar with 1 tsp cinnamon in a shallow bowl for coating.
Lay a wrapper like a diamond. Spoon filling near the lower corner.
Moisten edges with water. Fold sides over the filling, then roll tightly to seal. Repeat.
Heat oil in a deep pot to 350°F.
Fry rolls in batches until deeply golden, 2–3 minutes, turning once. Drain on a rack.
Toss warm rolls in cinnamon sugar. Serve hot with caramel for dipping.