Preheat oven to 350 degrees F (175 degrees C). Lightly grease an 8-inch square pan.
Bring a large pot of lightly salted water to a boil. Cook egg noodles in the boiling water, stirring occasionally until cooked through but firm to the bite, about 5 minutes. Drain.
Beat cream cheese and brown sugar together in a bowl using an electric mixer until smooth and creamy; beat in cottage cheese, egg, egg whites, vanilla extract, cinnamon, and nutmeg until well combined. Fold noodles and apples into cheese mixture by hand; spread into the prepared pan.
Bake in the preheated oven until kugel is lightly browned and firm, 45 to 50 minutes.
