Put the potatoes in a saucepan and cover with water. Bring to a boil over medium-high heat and cook until tender, 15 to 20 minutes. Drain the potatoes then mash well in a large mixing bowl.
Mix the yeast and water in with the potatoes. Stir in the flour and 1 tsp kosher salt until thoroughly combined and a dough is formed.
Lay the dough on a lightly floured work surface and knead for 3 to 4 minutes, adding more flour if necessary.
Form the dough into a smooth ball and place in a large bowl with 1 TBSP oil, flip the dough ball around in the bowl to cover in oil. Cover the dough in a damp towel and set in a warm place to rise for about 1 hour.
Spread 1 TBSP olive oil in the bottom an 18x13 inch rimmed baking sheet.
Punch down the dough and stretch onto the oiled baking sheet so the dough is about 1 inch away from the rim on all sides.
Press the olives and tomatoes, cut side down, all over the top of the dough.
Sprinkle the Parmesan cheese, oregano and about 1/4 tsp kosher salt over the top of the dough then drizzle on about 1 TBSP of olive oil.
Let the dough rest uncovered for 20 more minutes.
Preheat the oven to 450°F.
Bake in the preheated oven for 15 to 20 minutes or until golden brown and crispy.
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