Prepare the skillet. Preheat the oven to 350°F. Generously grease an 8-inch cast iron skillet with butter and set aside.
Make the brown sugar filling. In a small bowl, mix together ½ cup packed light brown sugar and 2 teaspoons ground cinnamon until well combined. Set aside.
Brown the butter. Add ½ cup (1 stick) unsalted butter to a light-colored saucepan or skillet over medium heat. Once melted, the butter will begin to foam and bubble vigorously—this is the water cooking out. Continue cooking, stirring or swirling the pan frequently. The bubbles will gradually become smaller and finer, and you'll start to see golden-brown flecks forming on the bottom of the pan. As soon as the butter smells nutty and the milk solids are golden brown, not dark brown or black, immediately remove it from the heat and pour it into a heat-safe bowl, making sure to scrape in all those flavorful browned bits. Let cool slightly while you prepare the other ingredients.
Cook the apples. Melt 2 tablespoons butter in a skillet over medium heat. Add 2 large apples (about 2 cups chopped)and sauté until they just begin to soften. Add ¼ cup granulated sugar and 1 teaspoon ground cinnamon and continue cooking, stirring occasionally, until the apples are tender and most of the juices in the pan have been absorbed. Remove from the heat and set aside.
Mix the dry ingredients. In a medium bowl, whisk together 1½ cups all-purpose flour, 1 teaspoon baking powder, 1 teaspoon baking soda, ½ teaspoon salt, and ½ teaspoon ground nutmeg. Set aside.
Make the cake batter. Add the slightly cooled browned butter, ½ cup packed light brown sugar, and ¼ cup granulated sugar to the bowl of a stand mixer. Beat for about 2-3 minutes, until thoroughly combined and slightly thickened. It will not become light and fluffy.
Finish the batter. Add the dry ingredients in two additions, alternating with ½ cup buttermilk*, mixing on low just until combined after each addition. Avoid overmixing.
Layer the cake. Spread half of the cake batter evenly into the prepared skillet. Spoon about ⅔ of the cooked apples over the batter, followed by ⅔ of the cinnamon-brown sugar filling. Spread the remaining batter over the top, then finish with the remaining apples and brown sugar filling.
Swirl. Run a skewer or thin knife through the cake horizontally and then vertically, creating swirls throughout the batter. Don't overdo it—you want distinct pockets of cinnamon sugar and apples.
Bake. Bake at 350°F for about 35 minutes, or until the center is set and a toothpick inserted into the cake portion comes out clean or with a few moist crumbs.
Brush with butter. Melt 1 tablespoon butter and brush it generously over the top of the cake as soon as it comes out of the oven. Let the cake cool completely in the skillet before glazing.
Make the glaze. Whisk together ¾ cup confectioners' sugar, 2 tablespoons milk, and ½ teaspoon vanilla extract until smooth. Add additional milk, up to 1 tablespoon, a little at a time until the glaze reaches your desired consistency.
Glaze and serve. Pour the glaze generously over the completely cooled apple fritter skillet cake. Let the glaze set slightly before slicing and serving.
