Preheat the oven to 500F and place a baking stone or pizza metal on the lowest rack in the oven.
Make cheese mix: combine the queso quesadilla and the monterey jack cheeses in a medium bowl. In a another small bowl mix the salsa verde, goat cheese, sour cream, cream, and cilantro. Pour into the shredded cheese bowl and gently combine until blended but chunky.
To make the pizza, sprinkle some flour over a clean surface and roll the pizza dough as thinly as possible into a random shape about 13-inches across and place on a floured pizza peel.
Top with salsa verde, cheese mix, poblanos, chorizo, and cilantro.
Carefully slide flatbread off of the peel and onto the preheated pizza stone of pizza metal.
Bake until crispy and brown around the edges and the top is bubbling.