2) Slowly whisk the warm cream into the caramel, then the blackberry puree and liqueur, and cook until smooth and slightly thickened, about 2 minutes. Remove from the heat and whisk in the vanilla essence. Leave to cool to room temperature.
For the French toast:
1) Preheat the oven to 180C/Gas 4. Whisk together the eggs, sugar and salt in a large baking dish until smooth. Add the pumpkin puree, ground spices, milk, 60ml of the double cream, the vanilla and rum and whisk until smooth.
2) Put 4 slices of the brioche in the custard mixture and leave to sit for 2 minutes. Carefully turn over the slices and leave until the bread is completely soaked through, about 2 more minutes.
3) While the bread is soaking, heat 1/2 the butter and 1/2 the oil in a large frying pan over a medium-high heat until the mixture begins to sizzle. Cook the 4 slices of the bread until golden brown on the bottom, about 2 to 3 minutes. Flip over and continue cooking until the other side is golden brown, about 2 minutes longer. Place on a baking sheet in the oven to keep warm. Wipe out the pan with paper towels and repeat the process with the remaining butter, oil and pieces of bread.
4) Combine the mascarpone and the remaining 60ml of double cream in a bowl and use a stand mixer or hand-held mixer to beat until fluffy and smooth. Scrape the mixture into a bowl and fold in a few tablespoons of the blackberry-caramel sauce. Put the remaining sauce into a small saucepan over a low heat, or put it in the microwave for a minute or so to just warm through.
5) Remove the French toast from the oven and arrange on a serving platter. Top 4 slices of the toast with a large dollop of the blackberry-caramel mascarpone and some of the fresh blackberries. Cover with the remaining 4 pieces of toast and drizzle each with some of the warm blackberry caramel sauce. Sprinkle with icing sugar, garnish with a fresh mint sprig and serve.