Dough

1. In a large bowl, whisk together the flours, sugar, and yeast. Pour in the ice water and mix thoroughly using a Danish dough whisk or wooden spoon. Add more water if necessary to make a fully-hydrated dough. Let the dough rest for 10 minutes.
2. Add the oil and the salt to the dough and mix until the dough forms a ball. Knead the dough in the bowl for a few minutes, then form it into a ball. Cover the bowl tightly with plastic wrap, and refrigerate for at least 24 hours, and up to 3 days.

Topping

1. Heat three tablespoons of the olive oil in a 12 inch non stick frying pan over medium heat. Add the onions, red peppers, sugar, garlic, salt, pepper flakes, and bay leaves. Cover the pan and cook for 10 minutes over medium low.
2. Remove the lid and continue to cook, stirring regularly, for another 10 to 15 minutes, until the onions are golden brown. Remove the pan from the heat and discard the bay leaves. Transfer the mixture to a heatproof bowl and stir in the vinegar. Cool completely before using. You can make the mixture in advance and refrigerate overnight.

To Make the Coques:

1. Deflate the dough and divide it into four equal sized pieces. Shape each piece into a tight ball and place, seam side down on your work surface, and cover with oiled plastic wrap. Let rest for an hour.
2. Place oven racks in the upper and lower third positions and preheat the oven to 500 degrees F. Brush two half sheet pans with 2 tablespoons of olive oil each.
3. Place one dough ball on your work surface, and roll it out to a 15 inch by 5 inch oval. Place it on the baking sheet, lengthwise. Repeat with the rest of the dough balls, two per baking sheet. If the dough springs back, let it rest for another 10 to 20 minutes, and spread it out with your fingers.
4. Dock each piece about 15 times with a fork. Brush each oblong piece of dough with the rest of the olive oil.
5. Bake the dough for 8 minutes, switching the pans at the four minute mark.
6. Remove the pans from the oven, and spread them with the red pepper and onion mixture. Sprinkle with the pine nuts. Place the baking sheets back into the oven, and bake for 16 minutes, switching and rotating the pans at the 8 minute mark. Continue to bake until the flatbreads are golden and crispy.
7. Remove the pans from the oven and let cool on the pans for about 5 minutes. Sprinkle with the parsley, and transfer to a cutting board to slice and serve.