Preheat the oven to 350°F. Lightly grease (or line with parchment) a large cookie sheet.
1) First layer: In a medium-sized mixing bowl, combine the butter, flour, and salt (if you're using it), working the butter into the flour 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.
2) Divide the dough in half; if you're using a scale, each half will weigh about 4 5/8 ounces. Wet your hands, and shape each piece of this wet dough into a rough log. Pat the logs into 10" x 3" rectangles on the 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.
3) Second layer: In a medium-sized saucepan, bring the water and butter to a boil. Stir until the butter melts, then add the flour (and salt, if you're using it) all at once. Stir the mixture with a spoon till it thickens, begins to steam, and leaves the sides of the pan; this will happen very quickly.
4) Transfer the dough 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.
5) Add the eggs one at a time, beating well after each addition; beat until the dough loses its "slimy" look, and each egg is totally absorbed.
6) Mix in the almond extract.
7) Divide the batter in half. Spread half the batter over one of the dough strips on the pan, covering it completely. Repeat with the remaining batter and dough.
8) With a spatula (or your wet fingers) spread the batter until it completely covers the entire bottom layer of dough. Smooth it out as best you can.
9) Bake the pastry for 50 minutes to 1 hour, or until it's a deep golden brown. Remove it from the oven, and transfer each pastry to a wire rack.
10) Topping: Spread each warm pastry with about 1/3 cup of jam or preserves. (Any flavor is fine, but our favorites are raspberry and apricot.)
11) 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.
12) Icing: Stir together the sugar, vanilla, and enough milk or water to form a thick but "drizzlable" icing.
13) Drizzle the icing atop the pastries.
14) Cut into squares or strips to serve.