Put the potatoes in a large saucepan and add cold water to cover by an inch. Bring to a boil, lower the heat and simmer until the potatoes are fork-tender, 5 to 8 minutes. Drain and mash into a creamy consistency; cover to keep warm.
In a large mixing bowl, dissolve the yeast in 1/4 cup warm water. Stir in the sugar and set aside until the mixture blooms with a slightly foamy top, 5 to 10 minutes.
Add the bread flour, semolina flour, mashed potatoes, olive oil, 4 teaspoons of the salt and 3 cups warm water to the large bowl with the yeast and mix with a wooden spoon until the ingredients come together into a soft, wet and uniform dough.
Transfer the dough to the bowl of a stand mixer fitted with a dough hook attachment. Knead on low speed, gradually increasing the speed to medium, until the dough is firm and has a smooth consistency, 8 to 9 minutes. Kneed in 1 to 2 additional tablespoons flour if the dough doesn't appear firm and smooth. Place the dough in a large greased bowl, cover with a dish towel and let rise at room temperature in a dark area until doubled in size, 1 to 1 1/2 hours.
Evenly grease a baking sheet with olive oil. Transfer the dough to the baking sheet and lightly stretch and press it towards the sides of the baking sheet until the dough is about 1/2 inch thick (it will not reach the sides). Let the dough sit for 10 minutes.
Continue to stretch the dough to the sides of the baking sheet using your fingertips; this time it will reach the sides. Cover the dough again and let it rise for 30 minutes.
Meanwhile, position an oven rack in the center of the oven and preheat to 400 degrees F.
Dimple the focaccia all over using your fingertips. Drizzle olive oil over the top of the dough and arrange the cherry tomatoes over the top cut-side down. Garnish with the oregano and remaining sea salt.
Bake until the focaccia is golden brown and a toothpick or knife inserted in the middle comes out clean, 40 to 50 minutes. Allow it to cool completely on the baking sheet.
Enjoy immediately or wrap it tightly and store on the countertop for up to 3 days.