If you don't have a food processor you can do this by hand. Sift flour, sugar and salt directly onto work surface. Tuck pads of in the flour. Press the butter into the flour, until crumbly, scraping and gathering together using a pastry scraper. Add cheddar to flour mixture, make a well in the center of the flour mixture and add water. Work together using pastry scraper until dough comes together into a ball. Flatten into a disk, wrap in plastic and chill for 30 minutes.
Combine the cream, star anise, ginger, cloves, cinnamon sticks, and nutmeg in a small saucepan and bring to a simmer. Remove from the heat and let steep for at least 20 minutes. Strain the mixture into a clean, small saucepan and place back over low heat while you make the caramel.
Combine the sugar, water and, vinegar in a medium saucepan over high heat and cook without stirring, until it's a deep amber color, about 8 minutes. Slowly whisk in the warm cream mixture a little at a time, and continue whisking until smooth. Pour 1/2 the caramel over apples; toss to coat. Set remaining caramel aside. Let stand until apples release juices, tossing occasionally, about 10 minutes.
Position rack in bottom third of oven and preheat to 375°F. Roll out dough on floured work surface to 16-inch round. Transfer to 9 1/2-inch-diameter spring form pan. Crimp edges decoratively. Spoon apple mixture into prepared crust. Sprinkle streusel over pie.
Bake pie until apples are tender and streusel is golden, covering crust edge with foil if browning too quickly, about 1 hour 10 minutes. Transfer pie to rack and cool at least 1 hour. Serve pie warm or at room temperature.
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