Peel the onions, cut in half and then cut into thin slices. Heat the olive oil in a saucepan and sauté the onions on a low heat for about 15 minutes until translucent.
Meanwhile, cut the puff pastry into 4 equal pieces. Lay out parchment paper on a baking sheet and place the 4 pieces of puff pastry on top.
Spread the onions on the tarts and place 2 slices of goat cheese on each. Pluck a few leaves of thyme from the stem and sprinkle over them.
Bake on the middle rack at 180 degrees for 8-10 minutes.