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.
Focaccia is perfect for buffets and dinner parties. You can prepare it in advance, up to 24 hours, and keep it out of the fridge covered with foil. Warm in the oven before serving, if possible, and it will be perfect!
Per Serving: 386 calories;12.3 g fat;
58.9 g carbohydrates;
10 g protein;
0 mg cholesterol;
1296 mg sodium.