Place the unsalted butter in a large cast iron or nonstick skillet. Set the skillet over medium heat and allow the butter to melt completely, being careful not to let it brown.
Once the butter is melted, add the white sugar, ground cinnamon, and ground nutmeg to the skillet. Stir the mixture until the sugar dissolves and the spices are fully incorporated, creating a fragrant mixture.
Add the peeled, cored, and sliced Granny Smith apples to the skillet with the butter, sugar, and spice mixture. Toss to coat the apples thoroughly, then cook, stirring occasionally, until the apples are soft and coated in the syrupy sauce, about 6–8 minutes. I like to check the apples frequently to make sure they become tender without losing their shape.
Once the apples are tender and coated in the spiced syrup, remove the skillet from the heat. Serve the saucy, spiced apples immediately while they are warm. Enjoy!