Piadina with Tomato, Burrata, & Arugula

Combine two tablespoons of olive oil and water in a measuring cup. Put the flour, salt, sugar, and yeast in the bowl of a food processor or a stand mixer with a dough-hook attachment. With the machine running, add the liquid and let the machine run until a sticky ball forms.
Wipe a bowl or deep container with olive oil. Put the dough in it and cover with a light kitchen towel. Set the dough in a warm place for two hours, or until doubled in size.
Remove the dough and divide it into four pieces. Form the pieces into balls, then flatten them into disks one-inch thick. Put the disks on a baking sheet, cover loosely with plastic wrap, and let set in a warm place for 15 minutes.
While the dough is on its first rise, preheat oven to 325°F. Line two baking sheets with aluminum foil.
Slice the tomatoes about half-inch thick. Lay the slices in one layer on the baking sheets. Top with the garlic and herbs. Sprinkle with salt and pepper and drizzle with olive oil. Roast for 20 minutes, rotating the baking sheets every five minutes, until they are soft but not falling apart.
To finish the piadina, heat a grill to medium-high.Shave the Parmesan with a vegetable peeler and set aside. Chop the burrata.
Flatten the disks of dough. Working on a lightly floured surface, roll each into a nine-inch round approximately one-fourth inch thick. Brush the disks with olive oil. Grill them on both sides for two and a half to three minutes, until the crust gets crispy.
Top the piadina with the burrata, tomatoes, and arugula. Sprinkle them with salt, pepper, and chili flakes. Drizzle with olive oil and top with the shaved Parmesan. Serve immediately.