Pre-heat your oven to 375.
In a large skillet or sauté pan, bring the oil to medium-high heat. Add the chicken breasts in a single layer and cook for 3 minutes on one side, flip them over, and then continue to cook for 3 minutes on the other side. Add the Buffalo sauce to the pan and wait until it bubbles (should happen almost immediately. Lower the heat to medium-low and cover the pan. Simmer for about 15-20 minutes (depending on the thickness of your chicken breasts) until the chicken is fully cooked through.
Remove the lid from the pan and transfer each breast to a cutting board, shredding it using two forks and then returning it to the sauce. When all the breasts are shredded, stir them into the sauce until thoroughly combined. Continue to cook for 3-5 more minutes until the sauce thickens and soaks into the chicken. Transfer the chicken to a mixing bowl and add the cream cheese and blue cheese dressing. Stir together until cream cheese melts in and the ingredients are well combined.
Unroll the pizza dough onto a piece of parchment paper and roll the dough to be 10”x14.” Using a pizza cutter or a sharp knife, make cuts up one of the longer sides of the dough, about 1 inch apart and about 3 inches long (into the center). Mirror those cuts on the opposite long side of the dough.
Spread the Buffalo chicken mixture down the center of the pizza dough, in a row with the cuts on either side. Starting at one end, fold a strip of the sliced dough on the side diagonally over the center ingredients. Fold the strip across from it over that strip in a criss-cross and repeat, alternating sides, until the strips of dough are covering the center in a braided pattern. Brush the top of the braid with the melted butter.
Transfer the parchment paper with the braid onto a baking sheet and bake in the oven for 20-25 minutes until the dough is golden. Slice into 12 equal pieces and serve (a serving is 2 pieces).