- Process flour, sugar, and yeast in food processor until combined, about 2 seconds. With processor running, slowly add ice water; process until dough is just combined and no dry flour remains, about 10 seconds. Let dough stand for 10 minutes.
- Add 1 tablespoon oil and salt to dough and process until dough forms satiny, sticky ball that clears sides of bowl, 30 to 60 seconds. Transfer dough to lightly oiled counter and knead until smooth, about 1 minute.
- Divide dough into 4 equal pieces (about 7 ounces each). Shape each piece into tight ball and transfer to lightly oiled baking sheet. Lightly brush or spray tops of dough balls with oil. Cover sheet tightly with plastic wrap. Let sit at room temperature until dough balls have relaxed and spread to 4 inches in diameter, about 11⁄2 hours. (Alternatively, unrested dough balls can be refrigerated—on sheet wrapped with plastic—for up to 3 days; let sit at room temperature until relaxed and spread to 4 inches in diameter, about 2 hours, before using).
- Pour remaining 2 tablespoons oil over dough balls. Coat both sides of 1 ball in oil and transfer to 12 by 16-inch sheet of parchment paper on counter. Using your fingertips and palms, gently press and stretch dough to form rough 13 by 9-inch oval of even thickness; transfer parchment with dough round onto second baking sheet. Repeat stretching dough on parchment with remaining 3 dough balls; stack stretched rounds between parchment on sheet. Press large sheet of plastic wrap directly onto surface of top dough round and let sit at room temperature while heating grill.
- Combine oil and rosemary in bowl. Whisk fig preserves and vinegar together in second small bowl. Set bowls aside.
- FOR A CHARCOAL GRILL: Open bottom vent completely. Light large chimney starter three-quarters filled with charcoal briquettes (41⁄2 quarts). When top coals are partially covered with ash, pour evenly over half of grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.FOR A GAS GRILL: Turn all burners to high; cover; and heat grill until hot, about 15 minutes. Turn all burners to medium-high.
- Clean and oil cooking grate. Grasp 1 long side of 1 dough round and parchment with both hands (gently holding dough and parchment together between your thumbs and fingers) and carefully flip onto grill, dough side down (over hotter side if using charcoal). Gently peel parchment away from dough with tongs, using second pair of tongs or spatula to secure dough in place on grill. Cook (covered if using gas) until bottom is well browned and spottily charred, 1 to 2 minutes, rotating dough once during cooking. Invert dough onto cutting board, grilled side up.
- Brush top of pizza with one-quarter of rosemary oil (2 heaping teaspoons). Sprinkle one-quarter of Manchego (about 3⁄4 cup) evenly over pizza. Drizzle one-quarter of fig mixture (1⁄3 cup) evenly over pizza. Using your hands or 2 sets of tongs, transfer pizza back to grill (over hotter side if using charcoal) and cook (covered if using gas), rotating pizza often until bottom is well browned and spottily charred and cheese is melted, 1 to 3 minutes. (If cooking over charcoal, middle of pizza will brown faster; to ensure even cooking, check bottom frequently, and rotate and reposition pizza as needed to ensure even browning from middle to edges.)
- Using 2 sets of tongs, transfer pizza to cutting board (if using gas, keep grill covered to conserve heat) and sprinkle with one-quarter of ham (about 3⁄4 ounce), one-quarter of basil (about 1 heaping tablespoon), and 1⁄4 teaspoon flake sea salt.
- Repeat with remaining dough and toppings, adjusting heat on gas grill as needed to ensure crust reaches visual cues in given time ranges for each pizza. Slice and serve pizzas immediately after removing from grill.
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