Preheat the oven to 475 degrees F and place a baking sheet or pizza stone inside to heat.
Roll or stretch the pizza dough into a 12-inch round on a piece of parchment paper.
Melt the butter in a large skillet over medium heat. Add the apple slices and brown sugar, then cook for 6 to 8 minutes, stirring occasionally, until the apples soften and the sugar caramelizes.
Remove the skillet from the heat and let the apples cool slightly.
Transfer the dough on the parchment to the hot baking sheet or stone. Arrange the brie slices evenly over the dough.
Spoon the caramelized apples and any pan juices over the cheese. Scatter the walnuts on top.
Bake for 12 to 15 minutes until the crust is golden and the cheese has melted.
Remove from the oven, sprinkle with fresh thyme leaves, and let rest for 2 minutes before slicing.