To Make the Ice Cream

In a bowl, whisk together the egg yolks, ¼ cup of the sugar, and the salt. Combine the milk, cream, vanilla bean and seeds, and the remaining ¼ cup sugar in a heavy saucepan. Cook over medium heat, stirring occasionally, until almost simmering. Slowly pour the liquid into the egg mixture, whisking as you pour.

Return the cream mixture to the saucepan. Cook over medium-low heat, stirring constantly with a heat-resistant plastic or wooden spatula, until the custard reaches 175°F and lightly coats the spatula. Strain the custard into a clean bowl, discarding the vanilla bean. Cool over an ice bath until room temperature. Refrigerate for at least 4 hours or up to overnight. Churn in an ice cream machine according to the manufacturer’s instructions. Put the ice cream in the freezer.

To Make the Cake

While the ice cream is freezing, preheat the oven to 350°F. Grease a 9½-inch springform pan and line the bottom with a round of parchment paper.

Sift together the flour and baking powder. Stir in the salt. In a medium bowl, cream the butter and sugar until light and fluffy. Add the eggs, one at a time, beating until well combined. Stir in the vanilla extract, crème fraîche or sour cream, and orange zest. Add the dry ingredients and mix until combined.

Spread the batter in the pan. Bake for about 20 minutes, or until a skewer inserted in the center comes out clean. Cool the cake completely.

Remove the cake from the pan by running a knife around the inside edge and releasing the latch. Place a cutting board on top of the cake and invert the cake and cutting board together. Remove the bottom of the pan and then carefully remove the parchment paper. Using a serrated knife, cut the cake in half horizontally. Wash and dry the springform pan. Place a cake layer in the bottom of the pan. Spread the vanilla ice cream over the cake. Put the half-assembled cake in the freezer.

To Make the Sorbet

Purée the raspberries and strawberries in a food processor. Strain the purée through a fine-mesh sieve into a bowl, discarding the seeds. There should be just under 2 cups purée. Stir the 1 cup sugar, the water, lemon juice, and salt into the purée. Taste for sweetness, adding more sugar as necessary. Refrigerate for at least 2 hours or up to overnight. Churn in an ice cream machine according to the manufacturer’s instructions. Spread the sorbet over the vanilla ice cream layer. Place the second cake layer on top of the sorbet, pressing gently to eliminate any air bubbles. Freeze until firm enough to slice, at least 2 hours or overnight, depending on your freezer.

In Advance

The cake can be assembled up to 2 days in advance. Wrap well in plastic wrap.