Makes one 15 1/2 inch by 10 1/2 inch rectangle or two 8 inch rounds
Note: Rapid-rise yeast reduces the preparation time by more than an hour. If you use an equal amount of regular active dry yeast instead, let the sponge develop for thirty minutes rather than the twenty, and increase the first and second rises to one and one-half hours each.
Boil 1 quart water in a small saucepan; add potato and simmer until tender, about 25 minutes. Drain potato well; cool until it can be handled comfortably and put through fine disk on a ricer or grate through large holes of a grater. Reserve 1 1/3 cups lightly packed potato.
Meanwhile, in a large bowl of electric mixer or workbowl of food processor fitted with steel blade, mix or pulse yeast, 1/2 cup flour, and 1/2 cup warm water until combined. Cover tightly with plastic wrap (or put workbowl lid on) and set aide until bubbly, about 20 minutes. Add remaining dough ingredients, including potato. If using a mixer, fit with paddle attachment and mix on low speed until dough comes together. Switch to dough hook attachment and increase speed to medium; continue kneading until the dough is smooth and elastic, about 5 minutes. For food processor, process until dough is smooth and elastic, about "40" seconds.
Transfer the dough (which will be sticky) to a lightly oiled bowl, turn to coat with oil, and cover tightly with plastic wrap. Let rise in warm, draft-free area until dough is puffy and doubled in volume, about 1 hour.
With wet hands (to prevent sticking) press dough flat into a generously oiled 15 1/2 inch by 10 1/2 inch jelly roll pan. Or, halve the dough and flatten each piece of dough into an 8 inch round on large, generously oiled baking sheet. Cover the dough with lightly greased or oil-sprayed plastic wrap; let rise in warm draft-free area until dough is puffy and doubled in volume, 45 minutes to 1 hour.
Meanwhile, adjust the oven rack to lower-middle position and preheat oven to 425°. With two wet fingers, dimple the risen dough at regular intervals.
Drizzle dough with oil and sprinkle evenly with rosemary and coarse salt. Bake until focaccia bottom(s) are golden brown and crisp, 23-25 minutes. Transfer to wire rack and cool slightly. Cut rectangular focaccia into squares or round focaccia into wedges. Serve warm. (Focaccia can be kept at room temperature for several hours and reheated prior to serving. Or, wrap cooled focaccia in plastic, then foil and freeze for up to 1 month. Unwrap and defrost in 325 degree oven until soft, about 15 minutes).
Focaccia with Black Olives and Thyme: Follow the recipe for Rosemary Focaccia, substituting 1 tsp fresh thyme leaves and 24 pitted large black olives (one per dimple) for the rosemary.
Parmesan Focaccia: Follow the recipe for Rosemary Focaccia, substituting 2/3 cup freshly grated Parmesan cheese for the rosemary and sea salt.
Sage Focaccia: Follow the recipe for Rosemary Focaccia, adding 1 TBS chopped fresh sage leaves with the other dough ingredients, and substituting 24 whole fresh sage leaves (one per dimple) for the rosemary.