Prepare the meat mixture first by finely dicing the carrot and the onion. Place the diced carrot and onion into a medium mixing bowl and add the beef mince, the egg, a splash of water, and the thyme leaves. Season with salt and pepper and mix well then set aside.
Place the butter puff pastry sheet onto a flat surface. Very lightly slice the puff pastry sheet in half lengthways. Divide the beef mince mixture into 2 portions. Using your hands roll each portion of the beef mince mixture into a sausage roll shape and place onto the centre of the pastry sheets lengthways. Fold over each side of the puff pastry and press down lightly then flip the sausage roll over. Slice off any excess pastry on the edges then slice into smaller sausage rolls. Brush with olive oil or egg. Transfer onto a paper lined baking tray.
Place the baking tray with sausage rolls into the oven and bake for 40 minutes. In the last 5 minutes of baking, brush with extra olive oil and sprinkle with sesame seeds.