In the bowl of a food processor outfitted with pastry blades, mix together 4 cups of the flour, the baking powder, salt, and sugar. Add the butter and pulse on and off until the mixture is mealy, 30 to 45 seconds. Alternatively, you can do this in a mixing bowl by hand, rubbing the flour mixture and butter together quickly with your fingertips.
In a medium bowl, whisk together the yogurt, olive oil, canola oil, and grated cheese. Add to the flour-butter mixture. If using a food processor, pulse it on and off until everything is combined, adding more flour 1 tablespoon at a time until the dough is dense and no longer sticky, being careful not to overwork. If it’s by hand, mix in the liquid with a spoon until a soft, sticky dough forms and continue to knead it until a dough mass forms, adding more flour incrementally as needed, to form a dense, heavy dough that doesn’t stick.
Shape the dough into a ball, cover with plastic wrap, and let it rest for at least 30 minutes in the refrigerator.
Remove the dough from the fridge and divide into 4 balls. Place in a floured bowl and cover with a kitchen towel.
Preheat the oven to 350°F and line 2 rimmed baking sheets with parchment paper.
Have a small bowl of cold water within reach. Take the first dough ball and roll it out on a floured surface to a circle about 14 inches in diameter. Using a 4-inch round cookie cutter, cut 8 to 10 circles out of the dough.
Using your index finger, rub the perimeter of half of them with a little water. Place 2½ to 3 tablespoons of the cheese filling in the center of each. Fit the remaining circles over the filling, flush and even with the bottom circles, pressing to seal the rim. You can press around the rim decoratively with the tines of a fork or carefully roll up the perimeter to form a decorative rim about ¼ inch wide. (It’s easier to use the fork!)
Place on a lined baking sheet and continue with the remaining dough balls and filling to make a total of 16 to 20 filled pies. If there is leftover pastry from cutting out the circles, put it all together and roll it out again to get a few more pieces out of the dough. Save a little filling for that, too.