Bring a large pot of water to boil and place the chicken breasts in the water, turn the heat down to low-medium, and let cook for 15 minutes before remove chicken breasts and letting sit for at least 10 minutes (make sure there is no more pink on the inside). Chop into small slices.
Meanwhile on medium-high saute separately the mushrooms in one pan and the onion/bell pepper/garlic in another pan. Once golden, combine in the larger pan and add the spices, chicken, and basil.
Preheat the oven to 425 degrees. Separate the dough into 5 even pieces and roll out each one into a thin oval. Place the filling in each one, add cheese, and fold over one side into the other, rolling the two sides together to seal the calzone.
Bake the calzones on aluminum-lined baking sheets for about 25 minutes or until golden on top.
Notes
You can try gently flipping them halfway to prevent burning on the bottom.