1 cup extra-virgin olive oil (from roasted garlic)
4 1/2 cups basil leaves (from 2 large bunches)
1/4 cup toasted pine nuts
1/4 cup finely grated Parmesan cheese
1/2 teaspoon lemon juice
Kosher salt and freshly ground pepper
For the Piadinas
All-purpose flour, for dusting
Oil, for grill
4 large or 8 small ripe tomatoes (about 1 pound), sliced
12 ounces (2 rounds) burrata or fresh mozzarella
20 medium basil leaves
Make the Dough: Combine flour and yeast in bowl of an electric mixer fitted with dough hook. Beating on low speed, add water. Keep beating until a shaggy dough forms, then beat 2 minutes more. Raise speed to medium and beat 2 minutes. Add salt and beat on medium-high, 2 minutes more.
Scrape into oiled bowl. (Dough will be quite sticky.) Cover with oiled plastic wrap and refrigerate overnight (at least 12 to 24 hours).
Roast the Garlic: Heat oven to 300 degrees. Cut off and discard top of garlic head. Place garlic in baking dish, cut side down, and cover with 1 cup extra-virgin olive oil. Cook until soft and golden, about 1 hour. Remove 6 cloves garlic (save rest for another use); reserve oil.
Make the Pesto: Bring a medium pot of water to a boil. Add basil and cook until bright green, about 30 seconds. Transfer to a bowl of ice water to stop cooking. Remove and squeeze dry. Combine in a blender with slow-roasted garlic, pine nuts, Parmesan cheese, and lemon juice. Blend to combine, drizzling oil from roasted garlic in through a feed tube. Season with 1/2 teaspoon salt and pepper.
Make the Piadinas: Heat grill to medium-high and oil.
Divide risen dough into 10 pieces and let rest for 30 minutes in refrigerator. Just before cooking, remove and let stand at room temperature for 10 minutes.
Dust dough balls with flour and stretch into 7-inch rounds.
Drop rounds onto heated grill. Cook 1 to 2 minutes. Turn and cook, until puffed and cooked through, 1 to 3 minutes more. Remove and cover with a kitchen towel (to keep pliable) while you cook the rest.
Spread a round with 2 tablespoons pesto. Top with a layer of tomatoes, 2 to 3 tablespoons burrata, and basil; season to taste with salt and pepper.