Line the bottom and over the sides of a 10-inch springform pan with parchment paper.
In a medium-size heavy skillet, combine 1/2 cup of the granulated sugar, the brown sugar, water, and 2 tablespoons of the Armagnac. Season with salt and pepper. Bring to a boil, reduce the heat to medium, and cook, stirring occasionally, until it reduces by half and caramelizes, 10 to 15 minutes. Remove from the heat.
Melt the butter in a medium-size sautĂ© pan over medium heat. Add the apples and cook until golden, about 5 minutes. Stir in the raisins and the remaining 2 tablespoons Armagnac and cook, stirring gently, for 2 minutes. Season with salt.
Spoon the apples and raisins into the spring-form pan and pour the caramel over the apples. Refrigerate for 30 minutes.
Preheat the oven to 350ÂşF. In a large mixing bowl, whisk the milk, eggs, the remaining 3/4 cup granulated sugar, the flour, and vanilla together until smooth. Strain the batter through a fine-mesh sieve over the apple-and-caramel mixture. Bake until the center sets, about 1 1/2 hours.
Remove from the oven and cool for 30 minutes. Then remove the sides from the pan, turn it over on a serving plate, and serve warm, garnished with the whipped cream, confectionersâ€™ sugar, and mint sprigs.
Yield: 16 servings