Drizzle a 9x13-inch baking pan with 2 tablespoons of the oil to coat the bottom evenly. Set aside.
Place the flour in the bowl of a stand mixer, fitted with the dough hook attachment. One one side of the bowl, add the yeast, and on the other side add the salt. Give it a stir to combine. Add in the remaining 3 tablespoons of olive oil and water and beat on high speed for 60 seconds.
Transfer the sticky batter into the prepared pan, cover loosely with plastic wrap and a damp kitchen towel, and let rise at room temperature for 1 hour. It should be puffy and full of air at this point.
While the dough is rising, preheat the oven to 375°F.
Once risen, use well-oiled fingers to gently poke holes all over the top of the dough. Drizzle the top lightly with 1 tablespoon of olive oil. Then top with the sliced plums, prosciutto, feta, and pistachios.
Bake until golden brown, 25 to 30 minutes. Remove from the oven and allow to cool for 5 minutes in the pan. Finish off with fresh rosemary and a sprinkle of salt and pepper. Cut into 12 squares and serve either warm or at room temperature. Leftovers can be wrapped in plastic or stored in an airtight container in the fridge for up to 3 days. Enjoy!