In a large stainless steel bowl, combine the all-purpose and rye flours, sugar, and salt and stir to mix well. I am a firm follower of the grating butter method. They recommend a pastry cutter, but I don’t own one, so I recommend grating the butter, using a cheese grater, into the dry ingredient mixture. Transfer to the freezer to chill for about 5 minutes. Mix the butter into the flour mixture, until throughly combined, breaking up the butter bits until they resemble the size of peas. The rest of the pie dough should look like canned Parmesan cheese. Speaking of cheese, now is the time to add the Cheddar and mix in quickly until evenly distributed.
At this point, add the water-vinegar mixture all at once, and give it a good mix. I like to use a spoon for this part. Scrape as much of the mixture as you can from one side of the bowl to the other, until you can’t see visible pools of liquid anymore. Now it’s hand time. Scoop up as much of the mixture as you can, and use the tips of your fingers (and a whole lot of pressure) to press it back down onto the rest of the ingredients. Rotate the bowl a quarter-turn and repeat. Scoop, press and turn. With each fold, your intentions is to be quickly forming the mixture into one cohesive mass.
Remove the dough from the bowl, place it on a lightly-floured counter, and use your bench scraper to divide into two equal pieces. Gently pat each into a 1-inch thick square, working quickly to seal any broken edges before wrapping them tightly in a double layer of plastic wrap. Refrigerate the dough for at least 2 hours, or, ideally, overnight.
You can keep the pie dough in the fridge for a few days or in the freezer for up to 1 year. If frozen, remove the dough and place tin the refrigerator to thaw one full day before you intend to use it.