In a large frying pan over medium heat, add the Olive Oil (1/4 cup). once the oil is heated up add your Yellow Onion (1 kilogram), and cook for about 3 minutes. Turn the temperature down to low and cook for 15 more minutes.
After about 20 minutes add the Brown Sugar (1/4 cup), and Balsamic Vinegar (1/4 cup) to the pan. Stir that around for about 5 minutes until everything is mixed well. After 5 minutes, allow your onions to cool while you work on the pastry.
Preheat oven to 400 degrees F (200 degrees C).
Using a sharp knife, cut your Puff Pastry (2) into quarters. Using the knife again, make a border of a centimeter thickness around the inside of each pastry piece.
Put the onion mixture on top of your pastry mixture making sure you stay within the borders you made. Sprinkle over some Cheddar Cheese (to taste) on top. Bake inside a preheated oven at 400 degrees F (200 degrees C) for about 15 minutes.