Combine flour, dry yeast, sugar and salt in a large mixing bowl and stir until blended.
Combine water and 2 tablespoons olive oil in a 2-cup glass measuring cup OR microwave-safe bowl. Microwave on HIGH in 15 second increments until very warm but not hot to the touch (120° to 130°F).
Add to flour mixture. Stir with a spoon until well mixed.
Drizzle 2 Tbsp olive oil into 9 x 13 pan and spread dough into pan. (Don’t worry if the dough isn’t even or filling the corners.) Cover with towel; let rise in warm place until doubled in size, about 30 minutes .
Push halved tomatoes into dough with cut side down.
Sprinkle with Parmesan cheese and sea salt.
Drizzle with remaining olive oil.
Let rise, uncovered, an additional 15 minutes while oven preheats to 375°F.
Bake for 25 to 30 minutes or until golden brown. Cool slightly. Cut and serve with extra oil for dipping if you’d like.