Combine 1/2 cup of the water, the sugar, and yeast in a large bowl. Set aside for 10 minutes.

 

Stir in 3/4 cup of the flour, cover, and let rise 2-1/2 hours.

 

While dough is rising, heat butter in a skillet over low heat. Add onion and sauté until onion is soft, but not browned, about 15 minutes.

 

Remove from heat and stir in basil, 1/2 tsp of the salt, and pepper. Add remaining flour to dough and beat well. Combine the 2 Tbsp olive oil and remaining warm water, then add to dough. Beat until dough forms a mass. Turn out onto a lightly floured surface and knead until dough is shiny and smooth, 8 to 10 minutes.

 

Transfer dough to a lightly oiled bowl and turn to coat all sides with oil. Cover and let rise until doubled in bulk, about 1-1/2 hours.

 

Preheat oven to 450 degree F. Punch dough down and roll into a 13 x 15 inch rectangle. Transfer to a baking sheet sprinkled with cornmeal. Spread top with onion mixture. Drizzle with additional olive oil. Sprinkle with the remaining coarse salt and bake until golden, about 15 minutes.

 

Cool slightly on a rack; serve warm.