Adjust the oven rack (with pizza stone) to the lower rack. Preheat the oven and pizza stone to 500 degrees F for at least 30 minutes.
Meanwhile, prepare some of the toppings: Heat 2 Tablespoons olive oil in a large sauté pan over medium-high heat. Scoop the flour into a shallow bowl. Coat the chicken breasts with flour and place in the heated pan. Sear for 2 to 3 minutes each side, lightly seasoning with salt, pepper and Italian seasonings. Continue to sauté the chicken until cooked through, while using your spatula to cut the chicken into 1-inch pieces. Transfer to a plate and set aside.
Lower the heat to medium and pour another Tablespoon or two of the olive oil into the hot pan. Add the shallots and sauté until just soft. Mix in the garlic and cook until fragrant, about 30 seconds. Deglaze the pan with the lemon juice and white wine. Cook for another 3-4 minutes to burn off the alcohol and absorb the liquid. Remove from heat and set aside.
Roll out the pizza dough. Brush the surface and outer crust with olive oil. Sprinkle with half of the shredded cheese. Layer with cooked shallots, chicken, sun-dried tomatoes, olives and capers. Top with remaining cheese.
Bake at 500 degrees F for 8-10 minutes, or until the crust is golden brown and cheese is melted and bubbly.