Preheat the oven to 400 degrees F and line a baking sheet with parchment paper.
Place the sliced carrots, beet, and parsnip in a bowl. Add the olive oil, thyme, salt, and pepper, then toss until the vegetables are evenly coated.
Unroll the pie crust onto the prepared baking sheet. Arrange the seasoned vegetables in the center, leaving a 2-inch border around the edges.
Sprinkle the goat cheese evenly over the vegetables.
Fold the crust edges over the filling, pleating as needed to create a rustic border. Brush the exposed crust with the beaten egg.
Bake for 30 to 35 minutes until the crust is golden and the vegetables are tender.
Remove from the oven and let the galette rest for 5 minutes before slicing.