Preheat oven to 450 degrees F. Line a baking sheet with parchment paper or spray with cooking spray.
Sauté the mushrooms in a skillet until the mushrooms give up their juice and are brown. Set aside.
In a medium bowl, combine the chicken and the pesto and toss to combine.
Divide the pizza dough in half and roll each section into a circle on a floured surface.
To assemble the calzones, spread half of the chicken onto the bottom 1/3 of each round of dough. Top each chicken mixture with the mushrooms and Swiss cheese. Take the top portion of dough and fold over. Press to seal.
Bake for 10-15 minutes or until the outside is golden brown. Slice and serve.