Bring a skillet to medium heat and drizzle with olive oil. Add chopped shallot, bell pepper and garlic and saute until peppers and shallots have softened, about 5 minutes.
If you haven't already, finely chop mushrooms in a food processor or you can chop by hand, just be sure there are no large pieces left. Add chopped mushrooms and stir everything together and saute for another 3-4 minutes. The mushrooms should have darkened in color slightly and softened.
Add chopped tomatoes and stir everything together. Continue cooking for another 4-5 minutes so tomatoes release their juices and have softened. Season with salt and pepper and give it one more stir.
Add the smoked paprika, tomato paste, pomegranate molasses, cumin and sumac and stir everything together.
Continue sauteing for another 2 minutes and the mixture reduces and thickens. Once doen, remove from heat and set aside.
Meanwhile, pre-heat oven to 500 degrees Fahrenheit and if you have a pizza stone, place that in the oven to get hot.
Lightly flour your work surface and pizza dough and divide dough into quarters. Roll each piece into a 10 inch diameter and as thing as possible.
Spoon a quarter of the mixture onto the rolled out dough, bringing it close to the edge in a thin layer. Use the back of a spoon to gently press the mixture into the dough. Continue with the rest of the mixture and dough.
Place pizzas on a baking stone and cook for 4-5 minutes until edges are crisp and lightly golden brown.
While pizzas cook, make the onion salad. Add red onion, sumac, olive oil, lemon juice and parsley to a bowl and toss to combine.
Once Turkish pizzas are done, remove from oven and top with sumac onion salad and serve.