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Heat the oven to 350°. Butter six 6-ounce ramekins or custard cups. Cover the bottom of each ramekin with a round of parchment or waxed paper. Butter the ramekins a second time.
In a large frying pan, melt 11/2 tablespoons of the cold butter over moderate heat. Add 5 tablespoons of the brown sugar and cook, stirring, until the mixture bubbles.
Add the apple slices and stir to coat with the sugar-and-butter mixture. Cook, stirring frequently, until the apples are softened and caramelized and almost all of the liquid has evaporated, about 8 minutes. Stir in a pinch of cinnamon and let cool.
In a small bowl, combine 1/4 cup of the flour, the rolled oats, 2 tablespoons of the brown sugar, 1/4 teaspoon cinnamon and a pinch of salt. Cut the remaining 2 tablespoons of cold butter into pieces and add them to the bowl. Using your fingers or a pastry blender, rub or cut the butter into the other ingredients until the mixture is the texture of coarse meal with a few pea-size pieces remaining.
For the batter, using an electric mixer, beat the almond paste with the granulated sugar until well blended. Add the 5 tablespoons of room-temperature butter and the shortening and beat until fluffy, about 5 minutes. Beat in the eggs one at a time. Add the vanilla and 1/8 teaspoon salt. With the mixer on low speed, add the remaining 3/4 cup flour and beat until blended.
Spread about 2 tablespoons of the batter in the bottom of each ramekin. Top with the apples. Add the remaining batter and smooth the tops. Sprinkle crumb topping over each and lightly press the crumbs into the batter. Bake until golden brown on top and the centers spring back when lightly pressed with a finger, 30 to 35 minutes.
Meanwhile, put the crème fraîche in a medium bowl. Stir in the remaining 1 1/2 tablespoons brown sugar. Leave until the sugar dissolves, about 10 minutes; then whisk the crème fraîche until it holds soft peaks.;
Run a knife around the edge of each ramekin and invert onto an individual plate. Lift off each ramekin so that the buckle slides out. Peel off the paper. Turn the desserts over so that the crumb topping is on top. Spoon some of the compote around each dessert and serve with a dollop of the brown-sugar crème fraîche.