Stir boiling water and cereal mix together in a medium bowl. Cover and let stand, stirring occasionally, until the mixture resembles thick porridge and cools to roughly 110°F, about 30 minutes.

Pulse the bread flour, whole-wheat flour, yeast, and salt in a food processor until combined. Dollop the porridge mixture over the top and drizzle in the vegetable oil. Process until a rough ball forms, 30 to 40 seconds. Let the dough rest for 2 minutes, then process for 30 seconds longer. If the dough is sticky and clings to the blade, add the remaining bread flour a tablespoon at a time and pulse to incorporate.

Turn the dough out onto a lightly floured counter and knead into a smooth, round ball. Place the dough in a large lightly oiled bowl and cover with greased plastic wrap. Let rise in a warm place until nearly doubled in size, 1 to 1.5 hours.

Adjust oven racks to the lower-middle and upper-middle positions and heat the oven to 350°F. Toss the tomatoes, zucchini, shallots, garlic, olive oil, balsamic vinegar, sugar, pepper flakes, salt, and pepper in a large bowl.

Spread the vegetables out over two rimmed baking sheets. Roast without stirring until the tomato skins have shriveled slightly but the tomatoes still retain their shape, 35 to 40 minutes, switching and rotating the sheets halfway through baking.

Place a baking stone on the lower-middle rack and increase the oven temperature to 500°F. Let the baking stone heat for at least 30 minutes to ensure a crisp bottom crust.

Turn the dough out onto a lightly floured counter, divide into three equal pieces, and cover with greased plastic wrap. Working with one piece at a time, press and roll the dough into a 16 by 8-inch oval. Transfer the dough to a rimless baking sheet lined with parchment paper and reshape as needed.

Lightly brush the dough with the remaining olive oil. Scatter one-third of the roasted vegetables evenly over the dough and sprinkle with feta, leaving a half-inch border at the edges.

Slide the parchment paper and flatbread onto the hot baking stone. Bake until the edges are deeply browned and crisp, 8 to 13 minutes, rotating the flatbread halfway through baking.

Slide the parchment back onto the baking sheet to remove the flatbread from the oven. Transfer to a cutting board, sprinkle with fresh basil, and slice. Let the stone reheat for 5 minutes between baking subsequent flatbreads.