Preheat oven to 350°F (175°C). Have a 9-inch round cake pan ready; do not grease or line it.

Prepare the topping: In the bottom of your cake pan, evenly distribute the cubed butter across the bottom. Sprinkle the brown sugar evenly over the butter.

Arrange the halved figs tightly in a single layer over the sugar mixture. I suggest placing them cut-side up to showcase the interior while baking.

Prepare the batter: In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.

In a large bowl, cream together the softened butter and granulated sugar until light and fluffy (about 3 minutes). Beat in the eggs one at a time, followed by the vanilla extract.

Alternately add the dry ingredients and the wet ingredients (milk and sour cream) to the butter mixture, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.

Carefully spoon the cake batter evenly over the fig and sugar layer in the pan. Smooth the top gently with a spatula.

Bake for 50 to 55 minutes, or until a wooden skewer inserted into the center of the cake comes out clean. Let the cake cool in the pan on a wire rack for exactly 10 minutes.

To invert, place a serving plate or cooling rack over the cake pan. Wearing oven mitts, carefully and quickly flip the pan and plate over. Let the pan rest on the plate for 5 minutes; the residual heat will help release the caramel. Gently lift the pan off the cake.