- Preheat the oven to 325 ºF (160 ºC). For the pistachio fingers, line 2 baking trays with parchment paper. On one parchment piece, trace a circle that is 8 ½-inches (21 cm) across and on the second, trace 2 sets of parallel lines that are 3-inches (7.5 cm) apart. Flip both parchment pieces over so that the marker lines are visible, but will not transfer onto the batter.
- In a food processor, pulse the flour, pistachios, cornstarch and baking powder until the pistachios are finely ground. Set aside.
- Using electric beaters or in the bowl of a stand mixer fitted with the whip attachment, whip the egg whites on high speed until they are foamy, then gradually add the 3 Tbsp (36 g) of sugar and continue to whip until they hold a soft peak when the beaters are lifted. Set aside or transfer to a separate bowl.
- Whip the egg yolks with the remaining ½ cup (100 g) of sugar until they are thick, doubled in volume and hold a ribbon when the beaters are lifted. Fold the whites into the yolks in two additions, alternating with 2 additions of the flour mixture until evenly blended. Spoon the batter into a piping bag fitted with a large, plain tip. Pipe a spiral of batter within the circle traced onto the parchment, to fill it in (this will be the base to the charlotte), then pipe 3-inch (7.5 cm) ladyfingers of batter on the second baking tray, leaving an inch (2.5 cm) between each. Bake the lady fingers for 9-11 minutes and the circle cake base for 11-13 minutes. Keep an eye on the oven for the last few minutes, to see that the ladyfingers only lightly brown. Cool the ladyfingers on the paper. If preparing a day ahead, peel the ladyfingers and base off of the paper and store in an airtight container.
- The bases for both fillings can be prepared at the same time. For the Blackberry Bavarian, soften the gelatin in the cool water in a small bowl. Whisk the milk, 1/3 cup (70 g) of the sugar and the vanilla in a medium saucepan over medium heat and have the egg yolks whisked with the remaining 1/3 cup (70 g) of sugar in a small bowl nearby. Pour the milk slowly into the egg yolks while whisking the yolks until combined, then return the entire mixture back to the pot, stirring until the custard has thickened and coats the back of a spoon, about 3 minutes. Remove the pan from the heat and whisk in the softened gelatin until melted. Set aside to cool to room temperature (if it sets up while cooling, it can be re-warmed on low heat until it is fluid again.) Whip the cream until it holds a soft peak when the beaters are lifted and fold this into the Bavarian base in two additions until incorporated. Stir the blackberry puree and lime (or lemon) juice together and fold this into the Bavarian base. Pour this into the pan, spreading to level, if needed. Chill if the white chocolate mousse layer isn’t ready yet.
- For the white chocolate mousse, soften the gelatin in the cold milk for 2-3 minutes, then warm the milk to melt the gelatin and set aside, keeping warm. Melt the white chocolate in a metal bowl placed over a pot of barely simmering water, stirring until smooth. Remove the bowl from the heat and whisk in the milk until smooth. Set aside to cool to room temperature. Whip the cream until it holds a soft peak and fold into the white chocolate base in two additions until incorporated. Pour this carefully over the blackberry Bavarian, spread to level and then chill uncovered for at least 5 hours.
- To finish the charlotte, remove the ring from the pan and transfer the cake to a cake stand or platter. Arrange the fresh blackberries around the edge of the charlotte and sprinkle the chopped pistachios in a line just inside the blackberries.
* Simple syrup is a combination of equal parts sugar and water brought to a simmer, until the sugar has dissolved and cooled. This can be made ahead and stored refrigerated for up to 2 weeks.