Soft on the inside, crispy on the outside, and sweet all around—these holiday desserts are surefire crowd-pleasers.
For the crust, in a food processor, pulse the flour, sugar, and salt until blended. Add the butter. Pulse until the butter is the size of chickpeas, about 6 times. In a small bowl, mix the ice water and vinegar. Sprinkle over the flour and butter. Pulse until moist clumps form. Transfer the dough to a work surface. Knead the dough a few times to bring it together. Divide into 4 portions. Roll the dough into balls; flatten into disks. Wrap the disks in plastic wrap. Refrigerate for 1 hour. (The dough can be refrigerated for up to 2 days or frozen for up to 2 months.)
For the topping, using your fingertips, in a medium bowl, mix 1/4 cup flour, 2 tbsp. sugar, and 1/8 tsp. cinnamon. Add the butter. Using your fingers, knead the butter into the dry ingredients until large clumps form. Add the halvah and toss. Refrigerate the topping while you make the crostatas.
Preheat the oven to 400°. Line a baking sheet with parchment paper. On a lightly floured surface, roll out 1 dough disk to a 6-inch round. Repeat with the remaining dough. Transfer the crusts to the baking sheet, spacing them evenly apart. (They may overlap now; that’s OK.) Refrigerate the dough until firm, about 30 minutes.
In a large bowl, whisk the remaining 1/3 cup sugar, 1 tbsp. flour, and 1/8 tsp. cinnamon and the cardamom. Mix in the apples and lemon juice. Dividing equally, mound the apples in the middle of the crusts. Fold the edges of the crusts up and over the edges of the apples, crimping the dough as you go around the crust. Top with the halvah crumb topping.
Bake the crostatas until the apples are tender and the crusts and topping are golden brown, tenting with foil if they brown too quickly, 25 to 30 minutes. Let the crostatas cool on a wire rack. Serve warm or at room temperature. Top with ice cream.
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