1. Pour 2 tablespoons olive oil into a 12-by-18-inch rimmed baking pan, and spread all over using fingertips; set aside. Place half the potatoes in a small saucepan. Cover with cold water; place over high heat. Add 3 teaspoons salt when water comes to a boil. Reduce heat to medium-high; cook until potatoes are tender, 10 to 12 minutes. Drain in a colander, reserving liquid. Pass through a potato ricer or a food mill and into a bowl; set aside.

2. Place 1/4 cup reserved warm liquid into the bowl of an electric mixer fitted with the paddle attachment. Sprinkle in yeast, and stir well; let stand until creamy, 5 to 10 minutes. Add 2 1/4 cups more reserved liquid, 2 tablespoons olive oil, and the reserved mashed potatoes; beat until combined. In a large bowl, whisk together flour and remaining 1 1/2 teaspoons salt; add to potato mixture. Mix on low speed until flour is incorporated, about 3 minutes. Change to the dough hook, and knead on medium-high until dough is smooth and elastic and is slightly tacky when squeezed but does not stick to fingers, 4 to 5 minutes.

3. Turn dough out onto a clean surface, and knead into a ball. Place in a lightly oiled large bowl, cover with plastic wrap, and let stand at room temperature until doubled in size, 1 to 1 1/2 hours.

4. Preheat oven to 425 degrees. Spread dough evenly in the prepared baking pan. Cover with oiled plastic wrap, and let stand in a warm place until dough has filled the entire pan and has increased in size by about one-third, about 30 minutes.

5. Using a mandoline or a knife, slice remaining potatoes into very thin rounds. Transfer to a bowl. Add half the rosemary and 1 tablespoon olive oil; season with salt and pepper. Toss to coat potatoes well.

6. Remove plastic wrap; dimple dough with fingertips, leaving deep indentations. Drizzle with remaining 2 tablespoons oil. Gently press reserved sliced potatoes into dough. Sprinkle with remaining rosemary. Bake until golden brown, 30 to 35 minutes. Remove from oven; transfer to a wire rack. Serve warm.