Place potato into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain and rinse under cold running water.

Peel potato; mash in a large bowl with a fork. Pour in flour and mix well with your fingertips. Rub together until mixture is even and free of lumps.

Combine 1 cup water, yeast, and sugar in a bowl. Mix well and let stand until frothy, about 5 minutes. Stir in 5 tablespoons olive oil.

Pour yeast mixture over flour mixture; mix, adding remaining 1 cup water a little at a time, until a soft dough forms. Mix in 1 tablespoon salt.

Knead dough on a clean work surface until smooth and soft, 8 to 10 minutes. Transfer dough to a greased bowl. Cover with a damp cloth; let rise in a warm place until doubled, about 2 hours.

Grease a large rimmed baking sheet with 2 tablespoons olive oil. Place dough on the baking sheet; stretch into the corners. Arrange tomatoes and olives over dough.

Mix remaining 1 tablespoon olive oil and 2 tablespoons water together in a bowl. Brush mixture over dough. Sprinkle remaining 1 tablespoon salt on top. Cover with a damp cloth; let rise for 30 minutes.

Preheat oven to 400 degrees F (200 degrees C).

Bake focaccia on the lower rack of the preheated oven for 15 minutes. Lift up with a fork to check bottom is golden brown. Transfer to the middle rack and bake until top is golden, 10 to 15 minutes more. Sprinkle oregano on top.