Preheat your oven to 350°F (175°C). Lightly grease a 9x13-inch baking pan with cooking spray to prevent sticking.
In a saucepan, combine the packed brown sugar, water or apple juice, cornstarch, lemon juice, vanilla extract, ground cinnamon, ground nutmeg, and salt. Stir well, then add the peeled and sliced apples. Cook over medium heat for 3-5 minutes, stirring frequently, until the apples start to soften and the sauce thickens slightly.
Pour the cooked apple mixture evenly into the prepared, greased baking pan, spreading the apples into an even layer.
In a large bowl, mix together the all-purpose flour, white sugar, baking powder, and salt. Stir in the milk and melted butter until just combined and no dry streaks remain. Avoid overmixing, as this will keep the topping tender. I like to use a spatula for gentle stirring to achieve a light, fluffy texture.
Pour the cobbler topping mixture over the apples in the pan, spreading it evenly. Sprinkle lightly with ground cinnamon. Bake in the preheated oven for 38–40 minutes, or until a toothpick inserted into the topping comes out clean and the top is golden brown.
Allow the cobbler to cool in the pan for at least 15 minutes before serving. Serve warm, ideally with a scoop of vanilla ice cream if you like. I find that a scoop of vanilla ice cream really brings out the flavors of the warm apples and spices. Store any leftovers covered in the refrigerator for up to 4 days.