Place the chopped potatoes in a medium saucepan and add enough cold water to cover them.
Bring to a boil and simmer until they soften, then drain and set aside to cool.
Combine the yeast, sugar, and 3/4 cup of warm water --- set aside until ready.
Mix flour, semolina, and salt in a large bowl.
Use a ricer to break the boiled potatoes and add them into the same bowl with semolina.
Pour the yeast mixture and mix everything until you get a dough-like consistency.
Transfer it to a flat surface and knead until you get a smooth and soft dough.
Oil the baking tray with about 2 tablespoons of olive oil --- coating your palms in this process.
Flatten the dough in the tray, using your hands to push and extend the focaccia Barese dough.
Cover with a clean kitchen towel and set aside until it doubles.
Preheat your oven to 200°C (375°F).
Add the halved cherry tomatoes, drizzle the remaining olive oil, and sprinkle some coarse salt and dried oregano.
Place the tray in the oven for 35 to 40 minutes or until it turns golden with crunchy edges.
Remove from the oven, transfer it carefully to a cooling rack, and set aside for a few minutes to cool.
Your focaccia Barese is now ready for serving!