Preheat oven to 350°F. Line bottom of an 8-inch round cake pan with parchment paper; grease pan with butter, and dust with flour. Set aside. Combine sugar, butter, and salt in a stand mixer fitted with the paddle attachment; beat on medium speed until light and fluffy, about 4 minutes. Add eggs, 1 at a time, beating well after each addition. Stir in vanilla and lemon zest. Add flour; beat on low speed until smooth, about 30 seconds. Pour batter into prepared pan; spread to evenly cover bottom. Arrange raspberries and strawberries on top of batter. Bake in preheated oven until a wooden pick inserted in center of cake comes out clean, 34 to 38 minutes. Let cake cool in pan 10 minutes. Run a small offset spatula around edge of pan to loosen cake; remove from pan. Let cool completely on a wire rack, about 1 hour. Dust with powdered sugar.