In a small bowl, dissolve the yeast in 1 cup (8 fl oz/250 ml) warm water (110°F/43°C), stir in the sugar, and let stand until foamy, about 5 minutes. Transfer the yeast mixture to a food processor. Add the olive oil, 2 Cups flour, and salt and pulse until a soft dough forms, adding more flour if needed to reduce stickiness. Turn the dough out onto a lightly floured work surface and knead until smooth and elastic, about 7 minutes. Form into a ball, transfer to a lightly oiled bowl, turn to coat, and cover the bowl with a damp kitchen towel or plastic wrap. Let the dough rise in a warm, draft-free place until doubled in bulk, about 1 hour.
Place a baking stone in the oven and preheat to 500°F (260°C). Dust a rimless baking sheet with the cornmeal. Divide the dough in half. On a lightly floured work surface, roll out half of the dough into a rough oval about 12 inches (30 cm) long and ¼ inch (6 mm) thick. Transfer to the prepared baking sheet. Arrange half of the tomatoes evenly over the dough. Sprinkle evenly with half each of the thyme and cheese, and dot with half of the olives. Slide the flatbread onto the hot stone and bake until the crust is bubbly and crisp on the bottom, 10–15 minutes.
Sprinkle with half of the oil, cut into pieces, and serve. Repeat with the remaining dough and toppings.