Preheat the oven to 325ºF. Line a large baking sheet with parchment paper.
Lay the sliced tomatoes in a single layer on the prepared baking sheet, then mist or drizzle lightly with olive oil and sprinkle with salt and pepper.
Bake for about 50 minutes to 1 hour, gently and carefully flipping the tomato slices once halfway during baking, until slightly shrunken and shriveled. Remove from the oven and let cool for about 5 minutes. Gently blot with paper towels to remove excess moisture, then transfer the tomatoes to a plate and set aside.
Increase the oven temperature to 500ºF. Re-line the large baking sheet with a clean piece of parchment paper and set aside.
For the mushrooms:
Meanwhile, add the olive oil to a large skillet set over medium heat. When hot, add the sliced mushrooms and cook until tender and they have released their juices, about 6 to 8 minutes. Remove from the heat and stir in salt and pepper to taste. Set aside.
On a floured surface, split the pizza dough into three equal pieces and shape each into a circle that is about 7 inches in diameter. If the dough keeps springing back, let rest for about 10 minutes and then try again.
On one half of each circle of pizza dough, add some of the sliced red onion, some crumbled goat cheese, the cooked mushrooms, the roasted tomatoes and some mozzarella. Fold the other half of the pizza dough over the top to cover the filling, pinch the two corners, but do not seal the dough all the way around.
Carefully transfer the assembled sandwiches to the prepared baking sheet. Bake until golden brown, about 10 to 12 minutes.
Remove from the oven and let cool for 10 minutes.
Meanwhile, in a small bowl, toss together the arugula, basil, lemon juice and olive oil. Carefully open each cooled sandwich and fill with some of the arugula mixture. Serve.