In a small saucepan, bring chiles and oil to a simmer. Remove from heat and let peppers steep in oil at least 1 hour.
To make the dough:
In a small bowl add the warm (this activates the yeast. If the water is too hot the yeast will die and if the water is too cold the yeast will not activate) tap water add the dry yeast and the sugar and stir to combine. Let sit for 15 minutes. The top of the water will bubble and it will smell very yeasty, this is what you want.
In a large mixing bowl, combine the flour and the salt. Make a well in the center of the flour and add the olive oil and the yeast/water mixture. Using a fork gradually stir the flour into the yeast/water mix. Once most of the flour has combined, turn the dough out to a smooth clean work surface and use your hands to knead the dough to a smooth consistency.
Return the dough to the mixing bowl, cover with plastic wrap and set in a warm place for at least 1 hour or until the dough has doubled in size.
Preheat oven to 425 degrees F and preheat grill to medium-high heat.
For the puttanesca topping:
In a small bowl, mix together the tomatoes, olives, capers, red pepper flakes and season with salt and a drizzle of olive oil, set aside.
Brush and oil grill to remove any crud and clean.
Portion the dough into golf ball size rounds. Using a rolling pin, roll each dough into a rectangle, I like irregular shapes. If not using the dough right away, wrap each 1 in plastic wrap and reserve until ready to use. If using the dough right away grill each dough on both sides so there are grill marks and the dough is stiff and crisp. Top some of the dough with shaved Parmigiano and the other with Taleggio.
Top the Taleggio pizzas with the tomato mixture. Place pizzas in oven until cheese melts, about 3 to 4 minutes. Immediately when the Parmesan pizzas come out of the oven, arrange prosciutto slices on top and then top with the arugula and drizzle with spicy chile oil. Cut pizzas into pieces and serve immediately garnished with chives.