Tomato Focaccia

Generously cover potato with salted cold water (1 teaspoon salt for 3 cups water) in a small heavy saucepan and simmer, uncovered, until just tender, 10 to 15 minutes. Drain and cool slightly, then mash until smooth.
Stir together warm water and sugar in bowl of mixer. Sprinkle yeast over mixture and let stand until foamy, about 5 minutes. (If mixture doesn't foam, start over with new yeast.)
Add potato and 1/4 cup oil to yeast and beat with paddle attachment at medium speed until combined, about 2 minutes. Remove paddle attachment and attach dough hook, then beat in 4 cups flour and 1 tablespoon sea salt at medium-high speed until combined well, about 3 minutes. (Dough will be very soft and sticky.)
Transfer to a well-floured surface and knead in remaining 1/4 cup flour with lightly floured hands until smooth and elastic, 8 to 10 minutes. (Dough will still be very soft and sticky.)
Scrape dough into a lightly oiled large bowl and cover bowl with oiled plastic wrap. Let dough rise in a draft-free place at warm room temperature until doubled, 2 to 2 1/2 hours. Generously oil a 15- by 10- by 1-inch baking pan.
Punch down dough (do not knead) and transfer to baking pan, then gently stretch to cover as much of bottom as possible (dough may not fit exactly).
Cover dough with oiled plastic wrap and a kitchen towel and let rise in a draft-free place at warm room temperature until doubled, 1 to 1 1/2 hours.
Preheat oven to 425°F with rack in lower third.
Arrange tomatoes on focaccia (do not overlap), then sprinkle with oregano and remaining 1/4 teaspoon sea salt and drizzle with remaining 1/4 cup oil.
Bake until center is firm, top is pale golden, and underside is golden (lift to check), 20 to 25 minutes.
Loosen focaccia from pan with a spatula and slide onto a rack to cool slightly. Cut into pieces and serve warm or at room temperature.