Preheat the oven to 350 degrees F and have a 10-inch oven-safe skillet ready.
In a large bowl, whisk together the brown sugar and eggs until smooth. Stir in the pumpkin puree, evaporated milk, pumpkin pie spice, salt, and vanilla extract until fully combined, then pour the mixture into the 10-inch oven-safe skillet.
In a small bowl, whisk together the all-purpose flour, sugar, baking powder, and salt to make the cake-style topping. Sprinkle this mixture evenly over the pumpkin layer, then drizzle the melted unsalted butter evenly over the top.
Bake the cobbler for 55 to 60 minutes, until the topping is slightly golden and set. Remove the skillet from the oven and let it sit and cool for at least 30 minutes so the filling can set; otherwise the interior will be quite gooey.
Serve slices with vanilla or cinnamon ice cream, a sprinkle of cinnamon or pumpkin pie spice, and chopped chocolate on top.