Preheat the oven to 350 degrees F. Put the whole, unpeeled potatoes in a pot with water to cover by a couple of inches, and bring to a simmer. Cook the potatoes until they are easily pierced with a knife but not falling apart, about 20 minutes. Let them cool slightly, then peel, press through a ricer onto a sheet pan, and let cool completely.
Scrape the potatoes into a bowl, and sift over them the flour, baking powder, and salt. Mix in the egg to make a smooth dough. Oil a 15-by-10-inch cookie sheet with olive oil. On a floured surface, roll the dough out to a rectangle almost as large as the cookie sheet, or as large as you can get it. (If the dough is difficult to roll, roll between layers of parchment paper.) Transfer to the cookie sheet, and press the dough out to almost reach the edges of the pan.
Top the dough evenly with the mozzarella, leing a 陆-inch border around the edges. Top with the sliced tomatoes, then drizzle with the 2 tablespoons olive oil. Sprinkle with the dried oregano and grated cheese. Bake until the crust is browned on the bottom and cooked through, about 35 minutes. Allow the pizza to cool on the baking sheet for a few minutes before cutting into squares to serve.