Step 1In a large, deep bowl or deep container (large enough to allow the dough to double or triple in size), whisk together the water, sugar and yeast. Whisk in the flour to form the biga, or starter dough; this will be a thin, sticky dough similar to wet cement. Cover the container and refrigerate for 3 days.
Step 1The day you plan to finish the focaccia, make the basil oil. In a blender, combine the basil, cheese and oil, and purée. Cover and refrigerate until ready to use.
Step 1In the bowl of a stand mixer, combine the yeast, sugar and water. Set aside until the yeast begins to bubble, or bloom, about 10 minutes.
Step 2Beat in the biga, then begin adding the all-purpose and bread flours, beating until incorporated to form a dough. Before adding the last cup of each flour, beat in the oil and salt. Add the rest of the flour, mixing until the dough is smooth and somewhat sticky. Remove the dough to a well-floured board and gently knead until the dough feels soft and smooth.
Step 3Place the dough in a greased bowl, cover and set aside in a warm place until the dough is risen and almost doubled in size, about 1 hour.
Step 4Heat the oven to 425 degrees.
Step 5Coat each of two (12-inch by 2-inch) cake pans generously with about 4 tablespoons olive oil and sprinkle the bottom of each pan with a light coating of cornmeal.
Step 6Divide the dough in half and form each half into a round. Place each round in a pan, gently stretching the dough to fit the diameter of the pan. If it springs back, give the dough a few minutes to relax, then stretch again.
Step 7Generously coat the top of each focaccia with a layer of basil oil, making sure the entire top is covered. Making a claw shape with your hands, press your fingertips into the dough, forming deep dimples in the dough and allowing the basil oil to penetrate into the holes. Set the focaccia dough aside until puffed and almost doubled in height, 20 to 40 minutes (rising timing will vary depending on the temperature of the room).
Step 8Coat the onion slices with the remaining basil oil (this will keep them from burning as they bake). Shake off the excess oil and sprinkle half of the slices over the risen dough, then sprinkle over ½ of the Parmesan cheese over the dough. Repeat with the second pan of dough.
Step 9Place one pan in the oven and bake until the focaccia is crisp and a rich golden-brown, and the cheese is toasted and onions are dark and crisp, 25 to 35 minutes. Check the focaccia occasionally as it bakes, checking to make sure it does not rise up over the sides of the pan and that no oil spills over. Rotate the bread about halfway through baking for even coloring.
Step 10Remove the pan to cool slightly on a wire rack, and bake the second focaccia. When cool enough to handle, remove the focaccia from the pan. The bread is best eaten within 12 hours. Store at room temperature. Warm, as desired, on a baking sheet in a warm oven.