Preheat oven to 350 degrees. Butter a 9-inch round cake pan and dust with granulated sugar, tapping out the excess.
Put goat cheese, sugar, vanilla, lemon zest and lemon juice in the bowl of a stand mixer fitted with the paddle attachment and beat until smooth. Stir in egg yolks 2 at a time, mixing well after each addition. Mix in flour until incorporated.
Put egg whites in another mixer bowl, fit the mixer with the whip attachment and whip on medium speed until soft peaks form. Using a spatula, gently fold egg whites into the goat cheese mixture. Spread the batter into the prepared pan. Bake until a skewer inserted into the center comes out clean, 25 to 30 minutes. Let cool to room temperature.
To serve, run a knife around the inside edge of the pan to loosen the cake. Invert a platter on top of the pan and then invert the plate and pan together. Lift off the pan. Top with berries and, if desired, dust with powdered sugar.
The cake may be made a day ahead and kept refrigerated. Place the berries on top of the cake just before serving.