Heat oil over low heat—1 tablespoon in the 8-inch skillet, 2 tablespoons in the 10-inch skillet, and 1/4 cup in the 12-inch skillet. While oil heats, grate potatoes, squeezing as much liquid as possible from them, either by hand or using a potato ricer.
A minute or 2 from cooking, increase heat to a strong medium. Add potatoes to the pan, pressing on them with metal spatula to form a flat cake. Cook until bottom of cake is golden brown and crisp, about 4 minutes for the 8-inch skillet, 7 to 8 minutes for the 10-inch skillet, and about 10 minutes for the 12-inch skillet.
Place a plate over the pan and invert the cake onto a plate. Add remaining oil to the pan—1 1/2 teaspoons for the 8-inch skillet, 1 tablespoon for the 10-inch skillet, and 2 tablespoons for the 12-inch skillet, sliding cake back into the pan. Continue to cook until golden brown on remaining side, about 4 minutes longer for the 8-inch skillet, 7 to 8 minutes for the 10-inch skillet, and 10 minutes for the 12-inch skillet.
Slide potato cake from the skillet onto a cutting board; sprinkle generously with salt and pepper. Leave 8-inch cake whole, halve the 10-inch cake, and quarter 12-inch cake. Serve with toppings of your choice.