Combine the yeast, the sugar, and 5 tablespoons of the warm water in the bottom of a large mixing bowl. Let stand for 6 to 8 minutes, or until the mixture is foamy like a head of beer.

Stir in the remaining water, the 2 teaspoons salt, and 2 teaspoons of the olive oil. Stir in the flour, 1/2 cup at a time, to form a dough that is stiff enough to come away from the sides of the bowl. Turn it out onto a lightly floured work surface. Wash and lightly oil the bowl. Knead the dough until smooth and elastic, 6 to 8 minutes, adding flour as necessary. Note: The dough can be mixed and kneaded in a food processor fitted with a plastic dough blade or an electric mixer fitted with a dough hook.

Place the dough in the oiled bowl, cover with plastic wrap and a dish towel, and let rise in a warm place until doubled in bulk, 1 to 2 hours. If using store-bought dried tomatoes, plump them in hot water (place in a bowl with hot water just to cover) for 15 minutes. If using homemade dried tomatoes, drain well.

Punch the dough down and roll it out into 1 large or 2 small ovals, each 1/2 inch thick. Transfer the oval(s) to a baker's peel (wooden baker's paddie), cookie sheet, or tart pan bottom liberally sprinkled with cornmeal. Cover the focaccia with a clean cloth and let rise until doubled in height (30 to 60 minutes). Place a baking stone or sheet pan in the oven on the middle shelf and preheat to 375° F.

Poke your fingers over the surface of the focaccia to decoratively dimple the surface. Arrange the tomato halves (cut side down) and basil leaves in an attractive pattern on top. Brush the top with 1 teaspoon olive oil and sprinkle with kosher salt and black peppercorns.

Slide the focaccia onto the baking stone and bake until crisp and golden brown, 20 to 30 minutes. Let cool slightly before serving. Just before serving, drizzle focaccia with the remaining olive oil. Cut into wedges and serve.