In a large bowl, whisk together the flour and salt until evenly combined.
In a small bowl, combine the warm water, yeast, and sugar. Stir gently and let sit for about 5 minutes until foamy.
Pour the yeast mixture and ¼ cup olive oil into the flour mixture. Stir with a wooden spoon until a shaggy dough forms.
Turn the dough out onto a lightly floured surface and knead for about 8 to 10 minutes, until smooth, elastic, and slightly tacky.
Lightly oil a clean bowl, place the dough inside, and turn to coat. Cover with a damp kitchen towel and let rise in a warm spot for about 1 hour, or until doubled in size.
Generously oil a rimmed baking dish. Transfer the dough to the pan and gently stretch it toward the edges. Do not force it; let it relax if needed. Cover again and let rise for 20 minutes.
Preheat the oven to 400°F.
Use your fingertips to dimple the dough all over. Drizzle generously with olive oil, then sprinkle with chopped rosemary and coarse sea salt. Gently press the cherry tomato halves and onion slices into the surface of the dough.
Bake for 20 to 25 minutes, until the top is golden and the edges are crisp. Remove from the oven and let cool slightly on a wire rack before slicing.