Special equipment: an 11-by-17-inch baking sheet
Put 1 1/2 cups warm water (about 110 degrees F) and 1 teaspoon of the sugar in a large bowl. Sprinkle the yeast over the top, and set aside until bubbly, 5 to 10 minutes.
Meanwhile, whisk together both flours and the kosher salt in a medium bowl.
Once the yeast is ready, add the oil to the bowl, then the flour mixture, and stir with a wooden spoon until a shaggy dough forms. Transfer the dough to a work surface dusted with all-purpose flour, and knead until it is smooth, elastic and slightly tacky, about 5 minutes.
Lightly oil the bowl that held the flour mixture, put the ball of dough in it, cover and let rise in a warm place until doubled in size, about 1 hour.
Generously oil an 11-by-17-inch baking sheet. Punch the dough down, and transfer it to the baking sheet. Gently stretch the dough and pat it to the edges of the baking sheet with your hands (it will recede slightly from the edges). (If the dough is resistant, let it rest for about 10 minutes, then try again.) Dimple the dough all over with your fingers. Scatter the tomatoes over the top. Loosely cover, and let rise in a warm place until almost doubled in size, about 30 minutes.
Meanwhile, preheat the oven to 425 degrees F. When the dough is ready, uncover it, generously drizzle with oil and sprinkle with sea salt. Bake until the bread is golden brown and the tomatoes are soft and juicy, 20 to 25 minutes. Sprinkle the Parmesan over the top as soon as the bread comes out of the oven. Let cool on the baking sheet on a cooling rack. Top with the basil, and serve warm or at room temperature.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.) Never leave a child unattended in the kitchen. Limit the child to tasks that are safe and age-appropriate.
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