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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 fitted with the flat beater attachment or working in a large bowl with an electric hand mixer, blend the almond paste, sugar, almond flour, and salt until the mixture is uniformly crumbly, about 5 minutes.
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Add the egg whites one at a time, mixing to make a sticky, smooth paste. Stir in the almond extract.
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To shape the almond horns: Place the sliced almonds in a wide, shallow bowl or pie pan. Using your fingers, gently crush some of the almonds to create a mix of larger and smaller pieces. Fill a small bowl with water and keep it nearby so you can lightly dampen your hands while shaping the cookies. (This will help prevent the dough from sticking.)
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Scoop a generous tablespoon of the soft, sticky dough into your wet palms; a slightly heaping Tablespoon Scoop works well here. Keeping your hands damp, gently roll and stretch the dough into a 4" log. Transfer the log to the sliced almonds.
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Dry your hands (otherwise the almonds will stick to your fingers) and roll the log of dough in the almonds to coat on all sides. Carefully transfer the almond-coated log to the prepared baking sheet and gently curve it into a U-shape. Press any loose almonds back onto the surface. Repeat with the remaining dough, rewetting and drying your hands as needed.
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To bake the almond horns: Bake for 15 to 20 minutes, rotating the baking sheets top to bottom and front to back halfway through, until they are puffed and starting to lightly brown around the edges. (The almond horns can also be baked one tray at a time if you prefer; you can shape and bake the first batch while you shape the second.) Remove them from the oven and cool on a wire rack.
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To dip in chocolate: In a small saucepan set over low heat or in the microwave, melt the chopped 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 a parchment-lined baking sheet and allow the chocolate to set, 30 to 60 minutes depending on the temperature of your kitchen.
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Storage information: Store almond horns in an airtight container at room temperature for up to 10 days; freeze for longer storage.
