Preheat the oven to 400 degrees F. Toss the beet cubes with olive oil, salt, and pepper on a baking sheet. Roast for 25 minutes until tender. Set aside to cool slightly.
While the beets roast, cut the puff pastry into four 4-inch circles and place them on a parchment-lined baking sheet. Prick each circle several times with a fork. Bake for 12 minutes until golden and puffed. Let them cool.
Melt the butter in a small skillet over medium heat. Add the walnuts and brown sugar. Stir for 3 to 4 minutes until the sugar melts and coats the nuts. Spread the candied walnuts on parchment to cool.
Spread a layer of softened goat cheese over each cooled pastry round. Top with roasted beets and a few candied walnuts.
Finish with fresh thyme leaves and a light drizzle of honey. Serve at room temperature.