1. In the bowl of a standing electric mixer fitted with the dough hook, combine the yeast with 2 tablespoons of the water and let stand until moistened. Add the flour, 1/3 cup of olive oil, sugar, 2 teaspoons of salt and the remaining water; mix at medium speed until a soft, supple dough forms, 10 minutes. Transfer the dough to an oiled bowl, cover with plastic wrap and let stand in a draft-free spot for 1 hour.
2. Position racks in the upper and lower thirds of the oven and preheat the oven to 400°. Line 3 large rimmed baking sheets with parchment paper. On one of the sheets, toss the potatoes with the rosemary and 1/2 cup of olive oil and season with salt. Spread the potatoes in a single layer and bake until tender, 15 minutes. Let cool.
3. Turn the dough out onto an oiled surface and press to deflate. Cut the dough into 12 pieces and roll into balls; transfer to the remaining 2 baking sheets and brush with the oil used to roast the potatoes. Let stand in a draft-free spot until nearly doubled in bulk, 1 hour. Fan 3 potato slices on the top of each roll and brush with more oil.
4. Set a sheet pan in the bottom of the oven and fill with water to create steam. Bake the focaccia rolls for 30 minutes, until golden brown; shift the pans halfway through baking. Transfer the pans to racks and let the rolls cool completely.