Melt butter in a small skillet over medium heat. Mix in brown sugar, cinnamon, salt, and nutmeg. Add apples; cook and stir until apples have softened, about 10 minutes. Set aside.

Bring water and rice to a boil in a medium saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and water has been absorbed, about 20 minutes.

Stir 1 1/2 cups milk into rice mixture; cook until hot, 2 to 3 minutes. Add brown sugar and stir until well combined. Cook over medium heat until thick and creamy, 15 to 20 minutes. Stir in cooked apples, remaining 1/2 cup milk, and beaten egg. Cook, stirring constantly, until heated through, about 2 minutes. Remove from heat and stir in butter and vanilla extract. Serve warm.