Step 1: Preheat your oven to 350°F (175°C). Place the cast iron skillet on the stove over low heat.
Step 2: Make the caramel base by adding 6 tablespoons unsalted butter and 3/4 cup packed dark brown sugar to the skillet. Stir gently as the butter melts and sugar dissolves, about 3-4 minutes. Remove from heat once combined.
Step 3: Arrange the peeled and sliced peaches over the caramel in a single layer, slightly overlapping but not crowded.
Step 4: In a large bowl, whisk together 1 cup all-purpose flour, 1 1/2 teaspoons baking powder, and 1/2 teaspoon salt.
Step 5: In a separate bowl, cream 1/2 cup unsalted butter (softened) and 1/2 cup granulated sugar until light and fluffy, about 2-3 minutes. Beat in 1 large egg and 1 teaspoon pure vanilla extract until smooth.
Step 6: Alternate adding the dry flour mixture and 1/2 cup whole milk to the creamed butter mixture in three batches, beginning and ending with flour. Mix gently until combined without overmixing.
Step 7: Pour the batter evenly over the peaches in the skillet, covering them completely without disturbing the arrangement.
Step 8: Bake in the preheated oven for 35-40 minutes, or until a toothpick inserted in the center comes out clean and the top is golden brown.
Step 9: While the cake bakes, prepare the salted caramel topping: melt 1/2 cup granulated sugar in a small saucepan over medium heat without stirring until amber. Add 3 tablespoons unsalted butter and stir until melted. Remove from heat and slowly whisk in 1/4 cup heavy cream. Stir in 1/2 teaspoon sea salt. Let cool slightly.
Step 10: Once the cake is done, let it cool for 10 minutes. Run a knife around the edges, then carefully invert the cake onto a serving plate so the caramel-glazed peaches are on top.
Step 11: Drizzle the salted caramel sauce over the warm cake. Serve slices with extra caramel on the side.