Trim the figs to remove the stem and cut in half lengthwise. Set aside.
Melt 4 Tablespoons of butter and pour into a 9x2-inch cake pan. Swirl the pan to coat evenly. Sprinkle the bottom of the pan with the brown sugar, then arrange the figs in the pan, cut-side down. Add as many figs as you can to the pan without overlapping.
In a small bowl, whisk together the flour, cornmeal, baking powder, and salt. Set aside.
In a large bowl, whisk together the sugar, sourdough discard, eggs, vegetable oil, yogurt and vanilla extract. Mix until smooth. Add the dry ingredients to the wet ingredients, whisking until smooth.
Pour the batter over the fig layer in the cake pan. Jiggle the cake pan until the batter is distributed evenly.
Bake for 45-50 minutes, until a toothpick inserted in the center of the cake comes out clean. Let the cake cool in the pan for 10 minutes, then run a butter knife around the edges of the pan. Place your serving plate on top of the cake in the pan (the plate will be upside down). Quickly flip the cake and the platter together so the cake is resting on the platter, right-side up. Carefully remove the cake pan (it will still be hot). You can now see your beautiful cake!
Let the cake cool fully (at least an hour) before slicing and serving.
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Notes
To make this recipe with active sourdough starter, replace the sourdough discard with active starter (200g). You will not need to make any other adjustments to the recipe.