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Preheat the oven to 325°F. Lightly grease two baking sheets or line with parchment.
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To make the dough: In the bowl of a stand mixer, blend the almond paste, sugar, salt, and almond flour until the mixture is uniformly crumbly.
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Add the egg whites one at a time, mixing to make a smooth paste. Stir in the almond extract.
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To shape the dough: Transfer the dough to a pastry bag fitted with a large plain tip. Pipe U-shaped horns about 2” long and 2” wide, leaving at least 2" of space between each horn.
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To add the almond garnish: Brush the horns with the egg white and water mixture and sprinkle with the sliced almonds. For best results, refrigerate the shaped cookies, uncovered, for at least 2 hours or as long as overnight.
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When ready to bake, preheat the oven to 325°F with a rack in the center.
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To bake the cookies: Bake the horns for 15 minutes, until they're starting to lightly brown around the edges. Remove them from the oven and cool on a rack.
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To dip in chocolate: In a small saucepan set over low heat or in the microwave, melt the chopped bittersweet chocolate with the vegetable oil or shortening, stirring until smooth.
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Working with one horn at a time, dip each end into the melted chocolate. Return the horns to the rack for the chocolate to set.
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Store cookies in an airtight container at room temperature for up to 10 days; freeze for longer storage.