Preheat the oven to 410 degrees F (210 degrees C). Grease a 12-inch pizza pan.

Place chicken into a large pot and cover with salted water; bring to a boil. Cook chicken in boiling water until no longer pink in the center and the juices run clear, about 15 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Transfer chicken to a bowl and shred with two forks.

Melt butter in a skillet over medium-high heat. Cook and stir shredded chicken and Buffalo wing sauce in hot butter until chicken is coated and heated through, 2 to 4 minutes.

Turn pizza dough out onto a generously floured surface. Roll dough into a 12-inch round. Brush dough with 1 tablespoon oil and sprinkle with 1 cup mozzarella cheese, leaving a 1-inch border. Spread chicken mixture over cheese; top with remaining 1 cup cheese.

Fold pizza dough in half and pinch edges together to seal. Transfer calzone to the prepared pan and brush the top with remaining 1 tablespoon oil.

Bake in the preheated oven until crust is golden brown, 17 to 20 minutes.