Assemble all your ingredients together.
In the bowl of a stand mixer with a dough hook attachment, combine the flour, baking soda, and salt. Add the butter and mix on low speed until well incorporated, approximately 2 minutes. While the mixer is running, slowly add the water one tablespoon at a time until the mixture forms a dough around the hook.
Transfer the dough to a lightly floured work surface and knead for 5 minutes until smooth. Cut the dough into 4 equal pieces and pat them into round disks. Cover with plastic wrap and refrigerate for 30 minutes.
Place a grill pan over medium-high heat or light a charcoal or gas grill. Dust a work surface lightly with flour. Roll out each piece of dough into an 8 to 10-inch circle, approximately1/8-inch thick. Brush each circle of dough with extra-virgin olive oil and grill for 3 to 4 minutes on each side. Remove the flatbreads from the grill and set them aside while you gather the ingredients for the filling.
Layer one-half of each flatbread as follows: mozzarella cheese, chicken, tomatoes, arugula, salt and pepper, and another layer of cheese.
Fold the flatbread over the filling and place back on the grill just to melt the cheese and warm the piadina through, approximately 2 minutes.
Wrap the piadina in parchment paper or napkins and eat them with your hands.