Serve Anna Olson's apple cakes warm with a dusting of icing sugar.
Preheat oven to 375 °F and lightly grease an 8-inch baking dish.
Peel apples a third down from the stem and generously core out apples with a melon baller, leaving about ¾-inch of apple and bottom intact, and reserving apple pieces that don’t have core or seeds. Place apples in prepared baking dish and brush tops with melted butter.
Puree apple pieces in a food processor and measure ¼ cup to use with sugar, then pulse in almonds, eggs, extracts and salt. Spoon mixture into the center of each apple and sprinkle sliced almonds on top. Bake for 45-50 minutes, until tops of apples are a rich golden brown.
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