To prepare the dough, place the warm water in the bowl of a stand mixer. Sprinkle the yeast and sugar on top and let bloom, about 10 minutes.
Add the flour, salt, and olive oil to the yeast mixture and turn the mixer (fitted with the dough hook) to medium. Knead until the dough comes together to form a smooth dough. Remove dough to an oiled bowl and place in the oven with the light turned on.
Let rise for one hour, or until doubled in bulk.
Heat a medium skill over medium heat and add 2 tablespoons of butter to the pan. Add the onion and peppers to the skillet and cook, stirring frequently, until soft and beginning to brown, about 5-10 minutes. Set aside.
When the dough has doubled, punch it down.
Preheat the oven to 500 degrees.
Roll the dough out on a well floured surface to a 12 inch circle and place on a pizza stone or pan.
Spread the salsa verde in a thin layer of the dough, adding more salsa verde if desired. Sprinkle with the cheese and top with the meat and vegetables.
Bake on the middle rack for 8-10 minutes or until the edges of the crust are golden brown and the cheese is melted.