With a large metal spoon, stir together the flour, salt, yeast, and water in a 4-quart bowl or the bowl of an electric mixer until combined. If mixing with an electric mixer, fit it with the paddle attachment and mix on low speed for about 2 minutes, or until all the ingredients are hydrated and begin to form a wet ball of dough. Let the dough rest for 5 minutes. Switch to the dough hook, add the olive oil, and resume mixing on medium-low speed for 3 to 4 minutes, or until all of the oil is incorporated and the dough is sticky, supple, and smooth; it should clear the sides of the bowl and stick just a little to the bottom. If the dough seems like a batter and does not have sufficient structure to hold itself together, mix in more flour by the tablespoonful. Even though it is sticky, the dough should still pass the windowpane test. If mixing by hand, repeatedly dip one of your hands or the spoon into cold water and use it much like a dough hook, working the dough vigorously as you rotate the bowl with your other hand. As all the flour is incorporated and the dough becomes a wet ball, about 3 minutes, stop mixing and let the dough rest for 5 minutes.
Then add the olive oil, dip your hand or spoon again in water, and continue to work the dough for another 3 to 4 minutes. The dough should be very sticky, but it should also have some texture and structure. If the dough seems like a batter and does not have sufficient structure to hold itself together, mix in more flour by the tablespoonful. Even though it is sticky, the dough should till pass the windowpane test.
Form the dough into a ball and place it in a bowl brushed with olive oil. Turn the dough to coat it with the oil, cover the bowl with plastic wrap, and immediately refrigerate it overnight. The next day the dough should have nearly doubled in size. Allow it to sit at room temperature for about 2 hours before making the focaccia.
In a bowl, whisk together all the ingredients. Let sit at room temperature for 2 hours before using.
Shape and dimple the dough in a 12 by 17-inch sheet pan using the 2 tablespoons olive oil for preparing the pan and the 1/4 cup olive oil for dimpling the dough. Let the dough rise at room temperature for 2 to 3 hours, or until it fills the pan.
While the dough is rising in the pan, prepare the potatoes. If using new potatoes, place them in a saucepan with water to cover, bring to a boil, and boil for about 10 minutes, or until they can be easily pierced with a fork. Drain, let cool (or plunge them in cold water to speed the process), and cut into 1/4-inch-thick slices. If using regular-sized Yukon Gold potatoes, slice them paper-thin using a food processor, a mandoline, or a chefs knife. In a bowl, combine the sliced potatoes, onion, and rosemary. Pour in the Herb Oil and toss gently to coat.
Preheat the oven to 500°F (260°C). When the focaccia is fully risen and ready to bake, remove the potatoes from the oil, shaking off the excess oil, and spread the slices over the surface of the dough, either randomly or stacked like dominoes. Place the sheet pan on the middle shelf of the oven, bake for 5 minutes, and then lower the temperature to 400°F (200°C). Bake for 15 minutes, then rotate the pan 180 degrees. Continue to bake for 20 to 25 minutes longer or until the dough and the potatoes are golden around the edges.
Remove the finished focaccia from the oven and immediately transfer it to a cooling rack. Drizzle any oil remaining in the pan, as well as any remaining Herb Oil, to taste, over the potatoes. Season with the salt and pepper, then let cool for at least 20 minutes before cutting and serving.