Preheat your oven to 475°F (245°C). Place a pizza stone or an inverted baking sheet in the oven while it heats so it gets very hot.
Lightly flour your work surface and stretch or roll the store-bought pizza dough into a roughly 12-inch circle or oval. Aim for an even thickness so it bakes uniformly.
Transfer the shaped dough to a piece of parchment paper for easy handling. Brush the top of the dough with 1 tablespoon of olive oil, going all the way to the edges.
In a small bowl, toss the shredded cooked chicken with the remaining 1 tablespoon olive oil, salt, and black pepper until the chicken is lightly coated and seasoned.
Spread the whole berry cranberry sauce evenly over the oiled pizza dough, leaving a small border around the edges for the crust.
Scatter the seasoned shredded chicken evenly over the cranberry sauce so each slice will get some chicken.
Arrange the sliced brie cheese over the chicken, spacing the wedges so they will melt into little pools across the pizza.
Sprinkle the shredded low-moisture mozzarella cheese evenly over the top to help hold everything together as it bakes.
Top the pizza with the thinly sliced red onion, distributing the slices so they can soften and caramelize slightly in the oven.
Carefully slide the parchment with the assembled pizza onto the preheated pizza stone or baking sheet in the oven.
Bake for 12 to 15 minutes, or until the crust is golden and crisp on the bottom, the brie and mozzarella are melted and bubbling, and the red onion edges are lightly browned.
Remove the pizza from the oven and let it rest for 3 to 5 minutes so the cheese can settle slightly.
Sprinkle the fresh chopped parsley evenly over the top of the pizza for a bright finish.
Slice the Cranberry Chicken And Brie Pizza into wedges and serve warm.