In a food processor combine 3½ cups flour, the yeast, sugar, and salt and pulse to blend. With the motor running, add the water and 2 tablespoons oil in a steady stream, and pulse until the dough comes together in a rough mass. Process for about 40 seconds more, remove the dough from the food processor (the dough will be very sticky), and knead into a smooth ball on a lightly floured surface, adding more flour if needed. Place in a large, oiled bowl and cover with plastic wrap. Allow the dough to rise in a warm place until it has doubled in size, about 1½ hours. Punch the dough down.

In a medium saucepan of simmering salted water, cook the potatoes until tender but not mushy, about 10 minutes. Drain and cool until easier to handle. Cut each potato in half, and then into ⅓-inch half-moons. Put the potatoes in a large bowl and toss with 4 teaspoons of the oil, the salt, pepper, and ½ teaspoon of the rosemary. Heat a heavy skillet over medium-high heat on the stove. Add the potatoes and cook until they are lightly browned, about 5 minutes, turning occasionally. Return the potatoes to the large bowl. In the same skillet, warm the remaining 2 teaspoons oil with the butter. Add the leeks and cook until softened and slightly golden, about 4 minutes, stirring occasionally. Put the leeks in the bowl with the potatoes and stir to combine.

Place the dough on a lightly floured surface and divide it into four equal pieces. Form into smooth balls, cover with a towel, and let rest for 10 minutes. Tear off four pieces of parchment paper and lightly brush one side with oil. Roll the balls out on the parchment paper into large, rough oval shapes a little less than ¼ inch thick, and keep covered for up to 20 minutes until it’s time to grill. (If the dough is resisting your efforts to roll it out nice and large, cover with a towel and let rest for 3 to 4 minutes. Then roll again.)

Prepare the grill for direct cooking over medium heat (350° to 400°F) .

Prepare the grill for cooking over medium heat (400°F) .

Brush one side of each piece of dough with oil. Place the dough on the cooking grate with the paper sides facing up. Grill over direct medium heat, with the lid closed, until the top of the dough bubbles and the underside is crisp and golden, about 3 minutes. Peel off and discard the parchment paper. Transfer the crusts to a work surface with the grilled sides facing up. Immediately spread each crust with equal amounts of the potato-leek mixture, the mozzarella and blue cheeses, and the remaining rosemary, leaving a ¼-inch border.

Brush one side of each piece of dough with oil. Place the dough on the cooking grate with the paper sides facing up. Grill over medium heat, with the lid closed, until the top of the dough bubbles and the underside is crisp and golden, about 3 minutes. Peel off and discard the parchment paper. Transfer the crusts to a work surface with the grilled sides facing up. Immediately spread each crust with equal amounts of the potato-leek mixture, the mozzarella and blue cheeses, and the remaining rosemary, leaving a ¼-inch border.

Return the flatbreads to the grill and cook over direct medium heat, with the lid closed, until the underside is nicely marked and the cheeses soften, 3 to 5 minutes. Remove from the grill and finish with a little olive oil, if desired. Cut and serve right away.

Return the flatbreads to the grill and cook over medium heat, with the lid closed, until the underside is nicely marked and the cheeses soften, 3 to 5 minutes. Remove from the grill and finish with a little olive oil, if desired. Cut and serve right away.