Blend water and potatoes until smooth, then put aside to cool to about 120 degrees, which is still very warm, but not hot. (Take care when whirring hot liquids in a blender; they can "erupt.") Or strain out potatoes, mash, then recombine with the water.
In a large bowl, whisk together flour, yeast, sugar and salt. Add the cooled potato purée and mix until you have a wet, sticky dough. This takes only about 30 seconds. Cover the bowl with plastic wrap and let it sit undisturbed until the dough triples in size, about 2 to 3 hours.
Lightly oil a rimmed baking sheet, or 2 (9- to 10-inch) round pans with 1 tablespoon olive oil. Scrape the dough into the pan. Dipping your fingers in remaining 1/4 cup olive oil, use your hands to gently pull and press dough to the pan edges. Drizzle remaining oil and use your fingertips to press down all over the dough to create dimples on the surface.
If desired, press toppings well into the dough in an attractive pattern.
Preheat oven to 400 degrees.
Let the dough rise in a warm, draft-free spot until it reaches the edges of the pan, about 45 minutes. Sprinkle with coarse salt, to taste.