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Preheat the oven to 350°F. Lightly grease a large baking sheet, or line it with parchment.
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To make the bottom layer: In a medium-sized mixing bowl, combine the flour and salt, then work in the cold butter with a pastry blender or fork, your fingers, or a mixer. Mix until everything is crumbly, then stir in the water. The dough will become cohesive, though not smooth.
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Divide the dough in half; if you're using a scale, each half will weigh about 145g. Wet your hands, and shape each piece of the soft dough into a rough log. Pat the logs into 10" x 3" rectangles on the baking sheet, leaving at least 4" (but preferably 6") between them, and 2" on each side. These puff up in the oven (hence the name), and you need to leave them room for expansion.
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To make the top layer: In a medium-sized saucepan, bring the water, salt, and butter to a boil. Make sure the butter is completely melted. Add the flour all at once. Stir the mixture with a spoon or heatproof spatula until it thickens, begins to steam, and leaves the sides of the pan; this will happen very quickly.
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Transfer the stiff batter to a mixing bowl, or the bowl of an electric mixer. Beat it at medium speed for 30 seconds to 1 minute, just to cool it down a bit.
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Add the eggs one at a time, beating well after each addition; beat until the batter loses its "slippery" look, and each egg is totally absorbed.
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Mix in the almond extract.
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Divide the batter in half. Portion half the batter onto one of the dough strips, half onto the other dough strip; if you have a scale, each half of batter will weigh about 269g to 284g.
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Working on one dough strip at a time, use a spatula (or your wet fingers) to spread the batter until it completely covers the dough, including its edges. Smooth out the top as best you can.
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Bake the pastries 65 to 70 minutes, or until they're a deep golden brown. Remove the pastries from the oven, and transfer them to a rack.
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To make the topping: If the jam or preserves are very thick, warm briefly in the microwave or on the stovetop. Spread each warm pastry with about 1/3 cup of the jam or preserves. Any flavor is fine, but our favorites are raspberry and apricot. For best results, use something that's fairly smooth, without overly large pieces of intact fruit.
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Sprinkle the toasted almonds atop the jam. By this time, your beautifully puffed pastries are probably starting to sink; don't worry, this is all part of the plan. Allow the pastries to cool completely, then drizzle with icing.
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To make the icing: Stir together the sugar, salt, extract, and enough milk or water to form a thick but pourable icing. Start with 1 teaspoon liquid; add more, bit by bit, only until the icing is thin enough to drizzle.
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Drizzle the icing atop the pastries.
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Cut into squares or strips to serve.
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Store at room temperature, lightly tented with plastic wrap, for a day or so; freeze for longer storage. Pastry is best served the same day it's made. If you plan on serving it the next day, rewarm briefly in a preheated 350°F oven, then add the jam, nuts, and icing just before serving.
