1 hour before cooking, remove dough from refrigerator. 30 minutes before cooking, move top oven rack to 4 inches from top of oven and preheat to 500°F (260°C). Place 1 large baking pan in oven to get really hot and lightly grease the other with spray or olive oil. Cut six 16-by-6-inch rectangles from parchment paper.
Meanwhile, press dough down in a bowl to deflate. Place dough on a clean countertop. Divide dough in half and then cut each into 3 equal pieces. Working with one ball at a time and keeping the others covered in slightly greased plastic wrap. Form a rough ball by stretching dough and your thumbs and pinching the edges together so top is smooth. Place completed balls 3-inches apart on the greased baking pan. Let rest covered in greased plastic wrap for 1 hour.
Taking one ball at a time, generously coat ball and counter with flour. Press and roll into a 14-by-5½-inch oval. Arrange on a parchment rectangle and reshape as needed. Place dough and it's parchment back on baking pan.
Brush oil over the top and spoon ⅙ of the topping and then ⅙ of cheese on top of each "canoe", leaving a ¾-inch border around the edges. Fold long edges over the filling to form a canoe shape. Pince ends together. Paint folded outer edges with oil.
Remove hot baking pan from oven and quickly slide 3 pide ontop its parchment rectangle onto pan at least 1-inch apart. Quickly return to oven and cook until the sides are a nice golden brown, 10 to 15 minutes. Separate from parchment and place on a wire rack to cool for 5 minutes. Repeat with other 3.
Serve: Sprinkle a pinch of mint over each and cut crossways into 2 or 3 pieces. Serve hot.