In a stand mixer fitted with the dough hook, mix the flour with the baking powder and 1 1/2 teaspoons of salt. Add the water, milk and butter and beat at low speed for 10 minutes. Form the dough into a ball and transfer to an oiled bowl. Cover and let stand for 3 hours.
On a floured surface, form the dough into 4 balls; set them on a floured baking sheet. Cover and let stand for 30 minutes.
Preheat the oven to 350°. Heat a large cast-iron skillet. Roll out 1 ball of dough to a 10-inch round; prick with a fork. Oil the skillet. Add the round and cook over moderately high heat until it browns on the bottom and bubbles on top. Flip the round and cook until it's browned on the bottom; transfer to a baking sheet. Repeat with the remaining dough.
In a bowl, toss the arugula and olive oil; season with salt and pepper. Divide the prosciutto, mozzarella and arugula among the flatbreads and fold over. Bake until the mozzarella is softened. Cut into wedges and serve.