Preheat the oven to 200°C. Chop the carrots and beetroot into small bite-size cubes. Slice the red onion and add them all to a lined baking tray, toss in olive oil, salt and balsamic vinegar, mix well. Bake for 25 minutes, until cooked through.
Chop the vegan feta into cubes, pick the thyme leaves. Cut the puff pastry sheet into quarters, approximately 18.5cm x 13cm each.
Using a rolling pin, roll each rectangle so that its size increases by 2cm on each side and transfer to two large baking trays lined with parchment, ensuring that each rectangle is well spread out from the other.
Remove vegetables from the oven and leave to cool. Spoon some of the vegetables into the middle of each of the 4 quarters of the puff pastry, leaving about 5cm uncovered all the way around so you can seal the parcels. Add a quarter of the vegan feta to each, then some of the thyme leaves.
Gather the sides of the pastry and pull them so that they overlap. Seal the parcel by pinching together and twisting slightly in the centre, taking care not to tear the pastry. Brush each parcel with oat drink.