Centre a rack in the oven and preheat the oven to 350oF. Butter the insides of six 2 ½ by 1 ½ inch high ring moulds, 4 ounce disposable aluminium cups, or 4-ounce muffing tins, dust them with flour, and tap out the excess.
Place the moulds or cups on a baking sheet (line the sheet with parchment if you’re using the ring moulds) and keep close at hand.
Sift the flour and baking powder together and reserve.
Put the egg whites and vanilla bean pulp in the bowl of a mixer fitted with the whisk attachment.
Beat the whites on low speed until they are broken up and foamy.
Increase the speed to medium-high an continue to whip until the whites form soft peaks. Gradually add the sugar and whip until the whites form firm, glossy peaks.
Spoon out a little of the whipped whites into a small bowl of a mixer fitted with the whisk attachment.
Using a large flexible rubber spatula, fold the flour into the whites in two to three additions.
When the flour is incorporated, fold in the egg white and butter mixture. Immediately spoon the batter into the prepared moulds.
Slide the baking sheet into the oven and bake for 8-10 minutes, or until a knife plunged into the centre of the cakes comes out clean. The cakes should be golden brown and springy to the touch, and they should pull away just a bit from the sides of the moulds, or unmould the cakes and transfer them to a rack to cool to room temperature.
Dice ¼ cup of the strawberries. Cut ¼ cup of the raspberries in half. Gently mix these berries with the blackberries and blueberries, and set aside.
Slice each of the remaining strawberries into 8 to 12 pieces, depending on their size, and toss the slices along with the remaining ¾ cup raspberries, the sugar, pectin and lime zest and juice into a medium saucepan.
Cut the vanilla bean lengthwise in half and, using the back of the knife, scrape the pulp out of the pod; toss the pulp and the pod into the pot.
Gently stir, then set aside for 5 minutes.
Place the saucepan over low heat and cook, stirring occasionally and always gently, just until the berries begin to render their juices, about 5 minutes.
Increase the heat to medium and continue to cook and stir until the raspberries lose their shape and the strawberry slices are soft but still intact, about 8 minutes.
Turn the marmalade into a bowl, press a piece of plastic wrap against the surface to create an airtight seal, and cool to room temperature.
When the marmalade is cool, remove and discard the vanilla bean and fold in the reserved fresh berries. (The marmalade can be made up to 2 days in advance and kept in an airtight container in the refrigerator. Use it at room temperature or slightly chilled.)
To Serve, for each portion, spoon a circle of marmalade onto the centre of a dinner plate. Place a cake on top of the marmalade, then, with a spoon, gently form a shallow hollow in the centre of the cake and fill with marmalade
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