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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
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For the gelato: Whisk together the sugar and egg yolks in a medium bowl until pale.
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Heat the milk with the vanilla bean and seeds in a medium saucepan over medium-high heat to steaming.
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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).
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Remove from the heat and chill completely.
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Pour the chilled base into an ice cream maker and spin until frozen.
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Meanwhile, in a medium bowl, stir the vegetable oil into the chocolate-hazelnut spread until well combined.
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Fold in the feuilletine.
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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.
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Reserve the remaining chocolate-hazelnut mixture for the cake.
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For the pan di spagna: Preheat the oven to 350 degrees F. Grease an 8-inch cake pan and line with parchment paper.
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Whip together the sugar and eggs in a large bowl with an electric mixer until airy, about 2 minutes.
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Sift the flour in batches directly into the egg mixture and fold in after each addition until combined.
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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.
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Let cool in the pan until cool enough to handle, then invert on a cooling rack to cool completely.
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For the diplomat cream: First, make the custard.
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Whisk together the egg yolks and half of the sugar in a medium heatproof bowl.
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Heat the milk, vanilla bean and seeds and remaining sugar in a medium saucepan until just boiling.
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Remove from the heat and whisk the milk mixture into the egg-sugar mixture.
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Then whisk in the flour.
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Return the mixture to the saucepan, bring to a boil and cook, whisking constantly, until very thick, 1 to 2 minutes.
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Remove from the heat, transfer to a shallow dish and cool completely.
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Whip the cream to medium peaks and fold into the cold custard to lighten it.
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Fill a pastry bag fitted with a large round tip with the diplomat cream.
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To assemble the cake: Place a baking sheet in the freezer to chill.
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Split the cake in half.
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Brush the bottom half with the orange liqueur using a pastry brush.
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Pipe the diplomat cream to cover the surface within 1/4 inch of the edge.
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Scatter the surface with the sliced strawberries, half of the raspberries and half of the blueberries.
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Top with the other half of the cake.
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Pipe the top with an even layer of the diplomat cream and smooth the surface with a small offset spatula.
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Put the white chocolate in a microwave-safe bowl.
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Microwave at 15- to 20-second intervals, stirring between each, until almost completely melted.
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Remove when you can still see pieces of the chocolate and continue to stir until the chocolate is completely smooth.
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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).
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Place back in the freezer for just a minute or 2 until set but still pliable.
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Once set, start at one end of a strip and use the edge of an offset spatula to lift and ease up the chocolate.
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Wrap around the edge of the cake.
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Repeat with the second strip so that the ends overlap.
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(Or see Cook's Notes for an easy white chocolate garnish alternative.)
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Cut the remaining strawberries in half.
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Garnish the top of the cake with the remaining raspberries and blueberries, the halved strawberries and the edible flowers if using.
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Slice and serve with a scoop of the gelato, and garnish with a spoonful of the reserved chocolate-hazelnut mixture.