Preheat oven to 350°F. Grease bottom and sides of an 8-inch springform pan or round cake pan with butter. If using a springform pan, line bottom with parchment paper, and grease parchment with butter. Dust pan and parchment with flour, tapping out excess. If using a round cake pan, make easy-lift handles out of parchment paper: Cut 2 strips of parchment that extend beyond the edges of the pan, and overlap them, forming an "X" shape. Place in bottom of buttered pan, top with parchment round, and grease paper with butter. Dust pan and parchment with flour, tapping out excess.

Sift together flour, cocoa, baking powder, and salt in a medium bowl; set aside. Beat granulated sugar and butter with a stand mixer fitted with the whisk attachment on medium-low speed until well combined, about 30 seconds. Add eggs, 1 at a time, beating well after each addition. Add vanilla. Increase speed to medium-high, and beat until batter lightens in color and is fluffy, about 2 minutes. Fold in flour mixture by hand until all dry ingredients are incorporated. Fold in chopped chocolate.

Pour batter into prepared pan, and smooth top using an offset spatula. Place 2 to 3 fig slices, slightly overlapping, in center of cake. Arrange remaining fig slices, slightly overlapping, in a circle around cake, about ½ inch from edge of pan. Bake cake in preheated oven until just set but still slightly jiggly in center, 34 to 38 minutes. Let cake cool completely in pan on a wire rack, about 2 hours. If using a springform pan, gently run an offset spatula around edge of pan, and carefully remove sides. If using a round cake pan, let cake cool completely before using handles to gently remove cake from pan. (Center will remain slightly soft even when cool.) Sprinkle with powdered sugar, and serve with unsweetened whipped cream.