These savory tarts are a specialty of Nice, in southern France. Traditionally the pissaladiere has a flaky crust, but this crisp crust gives it a lighter taste.
In a small bowl, combine warm water and yeast. Let stand 10 minutes until foamy. Brush two baking sheets with olive oil.
Place flour and salt in food processor. Add yeast mixture and 3 tablespoons oil; process just until dough comes together. Knead on lightly floured surface for 2 minutes. Divide in half. Shape each half into a log; divide each into 6 equal-size pieces; flatten slightly with palm to form disks. Transfer to baking sheets. Cover with plastic, then with hot, damp cloth. Let stand until doubled in size, about 1 hour.
Preheat oven to 400 degrees. Heat pizza stone for 30 minutes, or large baking sheet for 15.
In skillet over medium-low heat, heat 2 tablespoons oil. Add onions; cook until they begin to soften, stirring often, about 30 minutes. Add sugar; raise heat slightly. Cook until golden, stirring often, about 60 minutes more. Season with salt and pepper.
On lightly floured surface, gently roll or stretch each disk of dough into a 1/4- inch-thick oval. Brush with remaining 2 tablespoons oil; top with onions, eggs, fennel, olives, and Gruyere. Sprinkle with salt and pepper. Transfer to preheated pizza stone. Bake until crust is golden, about 12 minutes. Garnish with herbs; drizzle with oil, if desired. Cut into wedges; serve.