Special equipment: an instant-read thermometer, an ice cream maker, a pastry bag fitted with a large round tip, a pastry brush and a small offset spatula

For the gelato: Whisk together the sugar and egg yolks in a medium bowl until pale. Heat the milk with the vanilla bean and seeds in a medium saucepan over medium-high heat to steaming. Whisk a little of the hot milk into the egg-sugar mixture, then add the egg mixture back to the saucepan with the milk and cook, whisking, until it registers 180 degrees F on an instant-read thermometer (the mixture should coat the back of a spoon). Remove from the heat and chill completely.

Pour the chilled base into an ice cream maker and spin until frozen. 

Meanwhile, in a medium bowl, stir the vegetable oil into the chocolate-hazelnut spread until well combined. Fold in the feuilletine. 

When the gelato is finished spinning, fold in half of the chocolate-hazelnut mixture so it is swirly and place in the freezer to firm, 2 hours to overnight. Reserve the remaining chocolate-hazelnut mixture for the cake.

For the pan di spagna: Preheat the oven to 350 degrees F. Grease an 8-inch cake pan and line with parchment paper. 

Whip together the sugar and eggs in a large bowl with an electric mixer until airy, about 2 minutes. Sift the flour in batches directly into the egg mixture and fold in after each addition until combined. Fill the prepared cake pan and bake until a toothpick inserted in the center comes out clean and the top is golden and springs back when lightly touched, 25 to 30 minutes. Let cool in the pan until cool enough to handle, then invert on a cooling rack to cool completely.

For the diplomat cream: First, make the custard. Whisk together the egg yolks and half of the sugar in a medium heatproof bowl. Heat the milk, vanilla bean and seeds and remaining sugar in a medium saucepan until just boiling. Remove from the heat and whisk the milk mixture into the egg-sugar mixture. Then whisk in the flour. Return the mixture to the saucepan, bring to a boil and cook, whisking constantly, until very thick, 1 to 2 minutes. Remove from the heat, transfer to a shallow dish and cool completely. 

Whip the cream to medium peaks and fold into the cold custard to lighten it. Fill a pastry bag fitted with a large round tip with the diplomat cream. 

To assemble the cake: Place a baking sheet in the freezer to chill.

Split the cake in half. Brush the bottom half with the orange liqueur using a pastry brush. Pipe the diplomat cream to cover the surface within 1/4 inch of the edge. Scatter the surface with the sliced strawberries, half of the raspberries and half of the blueberries. Top with the other half of the cake. Pipe the top with an even layer of the diplomat cream and smooth the surface with a small offset spatula. 

Put the white chocolate in a microwave-safe bowl. Microwave at 15- to 20-second intervals, stirring between each, until almost completely melted. Remove when you can still see pieces of the chocolate and continue to stir until the chocolate is completely smooth. Remove the baking sheet from the freezer and spread 2 thin layers of the chocolate on it, each about 1/16 inch thick, 3 inches wide and 15 inches long (they can be slightly tapered at either end). Place back in the freezer for just a minute or 2 until set but still pliable. Once set, start at one end of a strip and use the edge of an offset spatula to lift and ease up the chocolate. Wrap around the edge of the cake. Repeat with the second strip so that the ends overlap. (Or see Cook's Notes for an easy white chocolate garnish alternative.)

Cut the remaining strawberries in half. Garnish the top of the cake with the remaining raspberries and blueberries, the halved strawberries and the edible flowers if using. Slice and serve with a scoop of the gelato, and garnish with a spoonful of the reserved chocolate-hazelnut mixture.