Preheat oven to 350°F.
Melt 3 Tbs. butter in a medium skillet over medium heat. Once the butter is foaming, add the shallots to the pan and season with a pinch of salt.
Lower the heat to medium low and slowly caramelize the shallots 30-40 minutes, until medium to dark brown.
Roll the pizza dough out on a floured work surface. Evenly sprinkle the cheese over the dough, followed by the shallots, torn prosciutto slices and half the chopped chives. Gently roll the dough from one end to the other, forming a log.
Slice into 1/2-inch rounds, and place on a rimmed baking sheet. Melt the last Tbs. of butter and lightly brush over the tops of the pinwheels.
Bake for 25 minutes or until the crust is golden brown and the cheese has melted and the ham is crisping up on top.
Transfer to a platter and garnish with the rest of the chives.