Preheat your oven to 350°F. While it heats, measure out all your ingredients and prepare your mise en place to streamline the baking process. Slice your peaches into 1/2-inch thick wedges and set aside. This upfront organization ensures you won't scramble for ingredients while mixing the batter.
In a 9-inch cast iron skillet or similar oven-safe pan, melt the butter over medium heat, then add the packed brown sugar and cook for 2-5 minutes, stirring occasionally, until the mixture becomes fragrant and slightly caramelized. Arrange the peach slices in a single layer over the caramel, pressing them gently into the mixture. Remove the skillet from heat and let it cool slightly while you prepare the batter—this prevents the peaches from cooking and becoming mushy.
In a medium bowl, whisk together the flour, baking powder, cinnamon, nutmeg, and salt. Set this mixture aside—keeping dry ingredients separate prevents lumps and ensures even distribution of the leavening agents throughout your cake.
In a small bowl, whisk together the room temperature milk and vanilla extract. Set aside. Using room temperature milk helps the batter emulsify properly and creates a more tender crumb.
In a large mixing bowl, beat the granulated sugar and melted (but cooled) coconut oil together for about 3 minutes until well combined and slightly lighter in color. Add the egg and beat until fully incorporated. Now, add half of the dry ingredient mixture from Step 3 and stir just until combined, then add the wet ingredient mixture from Step 4 and stir gently, and finally add the remaining dry mixture, stirring for about 1 minute total until just combined. I like to alternate between dry and wet ingredients this way because it prevents overmixing, which would develop gluten and make the cake tough.
Carefully pour the batter from Step 5 over the caramelized peaches in the skillet, spreading it gently and evenly to cover all the fruit. Place the skillet in your preheated 350°F oven and bake for 25-35 minutes, until the top is golden brown and a toothpick inserted into the cake portion (not the fruit) comes out clean or with just a few moist crumbs.
Remove the skillet from the oven and let it cool for about 15 minutes—this allows the cake to set slightly while the caramel remains pliable for easy release. Run a thin knife around the edges to loosen the cake, then carefully invert it onto a serving plate. If any peaches stick to the skillet, gently place them back on top of the cake. Serve while still warm to enjoy the cake at its best texture with the caramel sauce still slightly soft.