Preheat the oven to 325 F. Grease a 9-inch springform pan and line the bottom with parchment paper.
Toss the fresh figs with 2 tablespoons of brown sugar. Arrange the figs, cut side down, in the prepared springform pan.
In a saucepan, combine 6 tablespoons butter with remaining 1 1/2 cups brown sugar and bay leaves. Bring to a boil over medium-low heat. Immediately pour the hot caramel over the arranged figs in the pan.
In a large mixing bowl, beat the remaining 6 tablespoons softened unsalted butter until creamy and fluffy. Add in 1 1/3 cup granulated sugar, beating until well combined.
Add the eggs one at a time, beating well after each addition. Add the vanilla and beat until combined.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Add into the batter in alternating additions with the yogurt. Mix until just combined.
Carefully spread the cake batter over the caramel and fig layer in the springform pan.
Place the cake in the preheated oven and bake for about 60 minutes, or until a toothpick inserted into the center of the cake comes out clean or with a few moist crumbs.
Allow the cake to cool in the pan on a wire rack for about 10 minutes. Then, carefully run a knife around the edge of the pan to loosen the cake. Remove the sides of the springform pan.
Invert the cake onto a serving plate, so the fig and caramel layer is on top. Remove the parchment paper. Serve warm or at room temperature.