3 cups all-purpose flour
1¼ cups pastry flour
1 tablespoon sugar
1 tablespoon kosher salt
¾ teaspoon baking powder
1½ cups water
1¼ cups (2½ sticks) softened butter, divided
1. In a large bowl, whisk together the flour, pastry flour, sugar, salt and baking powder. Make a well in the center and slowly add the water, stirring with your hands or a wooden spoon, until a shaggy dough forms. Transfer the dough to a clean surface and knead until smooth and elastic, 5 minutes. Wipe the bowl clean and lightly grease with oil, then place the dough inside and cover with a kitchen towel to let rest 30 minutes.
2. Turn the dough onto a clean surface and, using a pastry scraper or large knife, cut into 9 even pieces. Cover and let rest another 30 minutes.
3. Grease one rimless metal baking sheet with 1 tablespoon of the butter. Line a second baking sheet with parchment and set aside.
4. Place one piece of dough onto the buttered baking sheet and pat 2 tablespoons of soft butter on top. Using flat hands, press and slide the buttery dough outward, stretching it into a very thin rectangle, around 10-by-12-inches. It's OK if a couple little tears occur.
5. Starting at one short end, fold up the dough until you have one long strip about 1 inch wide. Slightly flatten the strip and roll it up into a coil, then place onto the parchment-lined baking sheet, smooth-side up, resembling a snail shape. Cover with a kitchen towel and roll the remaining dough pieces. Refrigerate, covered, until chilled, about 1 hour.
6. Place one piece of dough in between 2 sheets of wax paper and, using a rolling pin, roll into a rustic 8-inch circle. Freeze the sheets until completely frozen, about 45 minutes. Do ahead: The dough can be rolled out into circles, frozen in between wax paper and stored in a large Ziploc bag for up to 1 month.
7. Prepare a grill for medium-high heat or set a large grill pan over medium-high heat. Add one frozen dough sheet, then immediately reduce the heat to medium and cook until the bottom is golden brown with large bubbles forming underneath the dough, 1½ to 2 minutes. Flip and cook another 1 to 1½ minutes until golden brown all over. Transfer to a plate and cover with a kitchen towel while baking the remaining dough sheets.