Step 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.

Step 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.

Step 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.

Step 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.