Preheat oven to 180°C and set pastry aside to defrost if using frozen. Gently heat oil and butter in a large frying pan over medium heat until butter has melted.
Add garlic and onion and cook until onions are slightly tender. Add beetroot and stir until evenly coated in butter.
Add balsamic vinegar, sugar and season to taste with salt and pepper. Cook for 15 minutes, stirring occasionally.
Spoon into a 20cm pie dish, making sure the beetroot are cut side up. Bake for 15 minutes.
Remove from oven and cover with puff pastry, tucking the edges into the sides of the pie dish to fit tightly around the filling. Brush with egg and bake for 20-25 minutes or until pastry is golden.
Leave to stand for 10 minutes before inverting onto a serving plate. Sprinkle with crumbled feta and rocket leaves. Serve.