For the flatbread:
1 1/4-ounce packet active dry yeast
Pinch of sugar
3 cups bread flour, plus more for dusting
Kosher salt
2 tablespoons extra-virgin olive oil, plus more for brushing
For the dressing:
3 tablespoons balsamic vinegar
3 teaspoons dijon mustard
1/4 teaspoon dried basil
Pinch of sugar
Kosher salt and freshly ground pepper
1/2 cup extra-virgin olive oil
For the sandwiches:
4 medium tomatoes, sliced (about 1 1/4 pounds)
1 1/2 pounds fresh mozzarella, sliced
16 to 20 fresh basil leaves
Kosher salt and freshly ground pepper
Make the flatbread: Stir the yeast, sugar and 1 1/2 cups warm water (about 110 degrees F) in a bowl. Let sit until foamy, about 5 minutes. Whisk the flour and 1 1/2 teaspoons salt in a large bowl. Add the yeast mixture and olive oil and mix with your hands into a smooth dough. Cover the bowl with plastic wrap; set aside in a warm place until the dough doubles in size, about 1 hour.
Turn the dough out onto a lightly floured surface and knead until slightly elastic, dusting with more flour if the dough is too sticky to handle, about 5 minutes. Return the dough to the bowl, cover and set aside again until doubled in size, about 1 hour.
Brush 2 baking sheets with olive oil. Turn the dough out onto a lightly floured surface and knead until smooth and elastic, dusting with more flour if needed. Cut the dough in half and form into 2 balls. Roll out each ball into a 12-by-3-inch rectangle, about 1/2 inch thick. Transfer each dough rectangle to one of the prepared baking sheets and let rest, uncovered, until a dry crust forms on top, 1 hour, 30 minutes to 2 hours.
Meanwhile, position racks in the middle and lower thirds of the oven. Place a pizza stone or inverted baking sheet on the bottom rack and preheat to 475 degrees F for at least 40 minutes.
Press your fingertips through the dry crust on top of the dough to create a spotted pattern. Gently stretch each piece of dough into a 14-by-4-inch rectangle, then brush the tops with olive oil and sprinkle with salt. Put 1 baking sheet of dough on the pizza stone and bake until the bottom of the bread is golden brown, about 7 minutes. Transfer to the middle rack and continue baking until golden all over, about 5 more minutes. Remove from the oven and immediately brush the top of the bread with olive oil and sprinkle with salt; let cool on the baking sheet on a rack. Repeat with the remaining dough.
Make the dressing: Whisk the vinegar, mustard and dried basil in a bowl. Add the sugar and 1/4 teaspoon each salt and pepper. Slowly whisk in the olive oil.
Assemble the sandwiches: Slice each flatbread in half horizontally and brush with the dressing. Top the bread bottoms with the tomatoes, mozzarella and basil leaves. Season with salt and pepper and cover with the bread tops. Cut into pieces.