1. For the sponge: In a large bowl, combine the sponge ingredients in the order listed, mixing until evenly moistened. Cover and let sit at room temperature for 1 hour.

  2. For the dough: Add the flour, salt, and olive oil to the sponge, kneading the dough until it's shiny and elastic.

  3. Place the dough in a greased bowl, cover, and let rise until doubled, about 1 1/2 hours.

  4. Preheat the oven to 400°F.

  5. For the topping: Spread the sliced onions on a parchment-lined baking sheet and drizzle with 1 tablespoon of the olive oil. Bake for 20 to 25 minutes, stirring every 10 minutes, until the onions are soft and beginning to caramelize.

  6. Mix the garlic, oregano, rosemary, salt, and remaining olive oil in a large bowl. Slice the potatoes thinly and toss them in the garlic-and-oil mixture. Drizzle a small amount of oil in a large, heavy skillet over medium-low heat.

  7. Layer the potatoes in the pan. Cover and cook for 5 minutes. Remove the lid, stir, and cook for 5 more minutes or until the potatoes are tender. Remove from the heat and let cool in the pan.

  8. To assemble: Lightly oil a 12" round pizza pan, either flat or deep-dish; or a large rimmed baking sheet (e.g., a half-sheet pan). Turn the dough out onto the pan. Grease your hands and pat the dough flat, pushing it to the edges of the 12" pan, or simply making a freeform oval or rectangle on the baking sheet. Spread the roasted onions on top and let the dough rise for 30 to 40 minutes, until puffy.

  9. Lower the oven to 375°F.

  10. When the dough has risen, transfer the potatoes onto the focaccia in an even layer. Bake for 25 to 30 minutes on a flat pizza pan, or 37 to 40 minutes in a deep-dish pan, until the crust is deep golden brown. Sprinkle with chopped fresh parsley (or basil) before serving.

  11. Store, covered, for 2 days at room temperature, or freeze for up to 3 months.