Preheat a pizza stone in a 500 degree F oven. Dust the top of a pizza peel with semolina flour, and lay the pizza dough on top.
Combine the olives, capers, tomatoes, garlic and red pepper flakes in a small bowl. Season with salt and a drizzle of olive oil.
Thinly spread the puttanesca sauce over the pizza dough and top with the taleggio.
Place the pizza on the pizza stone or a sheet tray and bake until the crust is brown and crispy and the cheese is melted, 7 to 10 minutes.
Towards the end of the cooking time, lightly dress the arugula with olive oil and a pinch salt. Top the finished pizza immediately with the arugula, cut into pieces and serve hot.
Pour the warm water into a small bowl (if the water is too hot the yeast will die and if the water is too cold the yeast will not activate). Stir in the sugar and dry yeast to combine and let sit 15 minutes. The top of the water will bubble and smell very yeasty, this is what you want.
Combine the flour and salt in a large mixing bowl. Make a well in the center and pour in the olive oil and yeast water. Gradually stir the flour into the liquid, using a fork. Once most of the flour has been incorporated, turn the dough out onto a smooth, clean work surface dusted with bench flour. Use your hands to knead the dough to a smooth consistency.
Place the dough in a lightly oiled mixing bowl. Cover with plastic wrap and set in a warm place until the dough has doubled in size, at least 1 hour 30 minutes.
Portion the dough and stretch to the desired size.