Preheat the oven to 350 degrees F. Butter just the sides of a 9-inch round cake pan.
For the caramel topping: Stir the sugar and 1/3 cup water in a medium saucepan over medium heat until the sugar dissolves. Stop stirring, and gently swirl the saucepan occasionally until the mixture is the color of light amber, 6 to 8 minutes. If needed, brush any sugar crystals off the side of the saucepan with a wet pastry brush. Remove the saucepan from the heat, and carefully whisk in the butter (take care: the mixture can bubble up considerably). Pour the caramel into the prepared cake pan, tilting the pan to cover most of the bottom (be careful: the pan will be very hot from the caramel). Arrange the figs, cut-side down, on the caramel.
For the cake: Spread the hazelnuts out on a baking sheet, and bake until nicely toasted, 8 to 10 minutes. Let cool, then pulse in a food processor until finely ground. Transfer to a large bowl and whisk them together with the flour, baking powder, baking soda and salt. Set aside.
Beat the butter and brown sugar with an electric mixer on medium-high speed in a large bowl until light and fluffy, about 4 minutes. Add the eggs 1 at a time, beating to incorporate after each addition and scraping down the side of the bowl as needed, then beat in the vanilla and lemon zest (it's okay if the batter looks slightly separated and broken). Adjust the mixer speed to low, and add 1/2 the flour mixture, then the sour cream, then the remaining flour mixture.
Pour the batter over the figs in the cake pan, spread it into an even layer and give the pan a few taps on the counter. Bake the cake until it's golden brown and pulls away from the edge of the pan and a toothpick or cake tester comes out clean when inserted in the center, 45 to 50 minutes. Run a knife around the edge of the cake to separate it from the pan. Let the cake cool in the pan on a wire rack until just cool enough to handle, 20 to 30 minutes. Invert a serving platter over the cake pan and, while holding them together, quickly flip them over so the cake unmolds, fig-side up. Let cool completely. Cut into wedges, and serve.
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