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Combine sugar and water in a small saucepan over high heat and cook, without stirring, until amber brown, about 8 minutes. Slowly whisk in the warm cream, blackberry puree and liqueur and cook until smooth and slightly thickened, about 2 minutes. Remove from the heat and whisk in the vanilla extract. Let come to room temperature.
Preheat the oven to 350 degrees F. Put a baking sheet in the oven.
Put 4 slices of the bread in the custard mixture and let sit for 2 minutes. Carefully turn over the slices and let sit until the bread is completely soaked through, about 2 more minutes.
While the bread is soaking, heat 3 tablespoons of the butter and 1 tablespoon of the oil in a large pan over 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 bottom is golden brown, about 2 minutes longer. Remove to the baking sheet in the oven to keep warm. Wipe out the pan with paper towels and repeat the process with the butter, oil and remaining 4 pieces of bread.
Combine the mascarpone and remaining 1/4 cup of heavy cream in a bowl and using a stand mixer or hand-held mixer, beat until fluffy and smooth. Scrape the mixture into a bowl and fold in a few tablespoons of the blackberry-caramel sauce. Put the sauce into a small saucepan over low heat and heat or put it in the microwave for a minute or so to just warm through.
Remove the French toast from the oven and arrange them on a serving platter. Top 4 slices of the toast with a large dollop of the blackberry-caramel mascarpone and some of the blackberries. Cover with the remaining 4 pieces of toast and drizzle each with some of the warm blackberry caramel sauce. Sprinkle with confectioners' sugar, garnish with fresh mint sprig and serve.