Note: Instant yeast is almost always sold under a marketing name; look for “rapid rise,” “perfect rise,” or “quick rise.” If your food processor includes a plastic dough blade attachment, use it; its short blades and dull edges make kneading easier on the motor. If not, the regular metal blade works almost as well. For best flavor, use high-quality oil-packed anchovies; in a recent tasting, Ortiz were our favorite. The dough in this recipe rises for 1 to 1 ½ hours. If a longer or overnight rise is more convenient, make the dough with ½ teaspoon of instant yeast and let it rise in the refrigerator for 16 to 24 hours. The caramelized onions can also be made a day ahead and refrigerated.
For the dough: In workbowl of food processor fitted with plastic dough blade (see note), pulse flour, yeast, and salt to combine, about five 1-second pulses. With machine running, slowly add oil, then water, through feed tube; continue to process until dough forms ball, about 15 seconds. Generously dust work surface with flour; using floured hands, transfer dough to work surface and knead lightly, shaping dough into ball. Lightly oil 1-quart measuring cup or small bowl, place dough in measuring cup, cover tightly with plastic wrap, and set aside in draft-free spot until doubled in volume, 1 to 1 1/2 hours.
For the caramelized onions: While dough is rising, heat oil in 12-inch nonstick skillet over high heat until shimmering but not smoking; stir in onions, salt, and brown sugar and cook, stirring frequently, until moisture released by onions has evaporated and onions begin to brown, about 10 minutes. Reduce heat to medium-low and cook, stirring frequently, until onions have softened and are medium golden brown, about 20 minutes longer. Off heat, stir in water; transfer to bowl and set aside. Adjust oven rack to lowest position, set baking stone on rack, and heat oven to 500 degrees.
To shape, top, and bake the dough: When dough has doubled, remove from measuring cup and divide into 2 equal pieces using dough scraper. Working with one piece at a time, form each piece into rough ball by gently pulling edges of dough together and pinching to seal (see illustration 1). With floured hands, turn dough ball seam-side down. Cupping dough with both hands, gently push dough in circular motion to form taut ball (see illustration 2). Repeat with second piece. Brush each lightly with oil, cover with plastic wrap, and let rest 10 minutes. Meanwhile, cut two 20-inch lengths parchment paper and set aside.
Coat fingers and palms of hands generously with oil. Using dough scraper, loosen 1 piece of dough from work surface. With well-oiled hands, hold dough aloft and gently stretch to 12-inch length (illustration 3). Place on parchment sheet and gently dimple surface of dough with fingertips (illustration 4). Using oiled palms, push and flatten dough into 14- by 8-inch oval (illustration 5). Brush dough with oil and sprinkle with 1/4 teaspoon pepper. Leaving 1/2-inch border around edge, sprinkle 1/4 cup olives, 1 tablespoon chopped anchovies, and 1 teaspoon thyme evenly over dough, then evenly scatter with half of onions (illustration 6). Arrange 6 whole anchovy fillets, if using, on tart and sprinkle with fennel seeds, if using. Slip parchment with tart onto pizza peel (or inverted rimless baking sheet), then slide onto hot baking stone. Bake until deep golden brown, 13 to 15 minutes. While first tart bakes, shape and top second tart.
Remove tart from oven with peel or pull parchment onto baking sheet; transfer tart to cutting board and slide parchment out from under tart. Cool 5 minutes; sprinkle with 1 1/2 teaspoons parsley, if using. Cut tart in half lengthwise, then cut crosswise to form 8 pieces; serve immediately. While first tart cools, bake second tart.
Note: Shaping Pissaladiere 1. Pull dough edges together.
Roll dough into taut ball.
Hold dough aloft and stretch.
Dimple dough with fingers.
Push dough into oval.