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To make the dough: Beat together the butter and sugar until light and creamy. This can be done by hand or in a stand mixer fitted with a paddle attachment on medium speed.
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Add the egg, vanilla, and salt and mix until thoroughly combined.
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Add the flour gradually, mixing until everything is completely combined.
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Remove the dough from the mixer and divide it in half. Press each half into a rough square and wrap with plastic wrap. Refrigerate the dough for 45 minutes (or for up to a week) before rolling and filling.
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To make the filling: Stir together the apricot preserves and bread crumbs.
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To roll, fill, and bake: Preheat the oven to 350°F. Lightly grease two baking sheets or line them with parchment paper.
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Remove one piece of dough from the refrigerator and unwrap it. Place the dough on a lightly floured work surface and tap it with your rolling pin to soften.
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Lightly flour the top of the dough, then gently roll it 1/8” thick.
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Using a 3" round cutter, cut out pieces of the rolled dough and place them on the prepared baking sheet. Continue rolling and cutting out circles, rerolling the dough as necessary.
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Place a rounded teaspoon of filling (15g to 20g) in the center of each dough circle. Brush the edges of the dough with water.
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Fold the edges of each cookie up and pinch together three corners to a make a triangle shape, with the filling visible in the center. Start with two sides, making an A shape; then fold in the third side to finish the triangle. Pinch the corners well and angle the sides towards the center, rather than vertically. You should strive for an equilateral triangle with edges that are about 1" long.
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Repeat with the remaining dough and filling.
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Bake the cookies for 18 to 22 minutes, or until the edges and bottoms are golden brown. There's a tendency to underbake these cookies; go for some color. It will give them a nice toasty flavor.
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Storage information: Store hamantaschen well-wrapped at room temperature for several days; freeze for longer storage.
