1. Make the dough: Pour warm water into a large bowl; sprinkle with yeast, and let stand until foamy, about 5 minutes. Whisk oil, sugar, and 2 teaspoons salt into yeast mixture. Add flour, and stir until a sticky dough forms. Turn out dough onto a floured surface, and knead until smooth, about 8 minutes. Transfer to an oiled bowl, and turn to coat. Let stand, covered, in a warm place until doubled in volume, about 1 hour.

2. Meanwhile, make the filling: Preheat oven to 400 degrees. Spread tomatoes on a rimmed baking sheet, drizzle with 3 tablespoons oil, and season with salt. Roast, flipping after 30 minutes, until shrunken and slightly dehydrated, 45 minutes to 1 hour. Let cool. Raise oven temperature to 500 degrees (with convection on if possible).

3. Turn out dough onto a lightly floured surface, and divide into 8 pieces. Form each into a ball. Let rest, draped with plastic wrap, for 15 minutes.

4. Place 1 dough ball on a lightly oiled baking sheet, and shape into a 7 1/2-inch round. Arrange a layer of tomatoes on half of the dough, and dot with 2 tablespoons goat cheese. Top with 4 slices soppressata. Tear off pieces of mozzarella, and place over top. Fold dough over to enclose filling, but do not seal. Repeat with remaining dough, tomatoes, goat cheese, soppressata, and mozzarella (arranging 4 sandwiches per baking sheet). Bake until golden and cooked through, about 10 minutes. Remove from oven. Let cool for at least 15 minutes before serving.

5. Meanwhile, toss arugula and basil with lemon juice and remaining 1/4 cup plus 2 tablespoons oil. Season with salt and pepper.

6. Open each sandwich, and fill with arugula-basil mixture. Cut each in half.