Whisk together the egg white, vinegar and 3 tablespoons water in a small bowl; set aside. Refrigerate the egg yolk in a small bowl, covered, until ready to use.
Pulse the flour, salt, mustard, paprika and turmeric in a food processor to combine. Add the butter, Cheddar and Parmesan, and pulse until the butter is completely broken up. Pour in the egg white mixture, and pulse until the dough comes together in a ball. (It's OK if it's a little wet.)
Place the dough on a large piece of plastic wrap, and pat it into a 1/2-inch-thick square. Wrap it up, and refrigerate to chill and firm up the dough, about 1 hour.
Position 2 oven racks at the top and bottom thirds of the oven, and preheat to 350 degrees F. Line 2 baking sheets with parchment.
Roll the dough out on a large piece of parchment (to keep it from sticking to the counter) into a 10-inch square about 1/8 inch thick. (Dust the dough with flour if you find it too sticky.) Trim the edges to straighten the square, then cut it into 1-inch squares (a pizza cutter works great).
Whisk 2 teaspoons water into the egg yolk. Brush the mixture over the tops of each cracker. Sprinkle each with the tiniest pinch of salt.
Bake the crackers, 2 baking sheets at a time, until they are deep golden brown on the bottom, about 20 minutes, rotating the sheets about halfway through. Let the crackers cool on the baking sheet for a few minutes, then transfer to a cooling rack to cool completely.
Store the crackers at room temperature in an airtight container for up to 2 days, or freeze them for up to 1 week.
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