Preheat the oven to 350 degrees F.

In a 12-inch saute pan over medium-high heat, melt the butter with the thyme and sugar. Crowd the onions into the pan, center cuts down and skins on. Allow the onions to caramelize in the pan over medium heat, approximately 5 to10 minutes. Place the pan in the oven and cook until the onions are fork tender, approximately 30 minutes. Remove from the heat and allow to cool slightly, approximately 15 minutes. Remove the tough, outside onion skins without disturbing the onions. Roll out the puff pastry on a well-floured surface. Use a fork to dock the dough. Whisk together the egg and water with a fork to create an egg wash. Using a pastry brush, brush the puff pastry with the egg wash. Carefully lift and place the puff pastry, egg wash side down, over the onions. Brush the exposed side of the puff pastry with the egg wash and begin to tuck the puff pastry into the circumference of the saute pan making sure to envelope all the onions, while lifting the corners of the puff pastry. Fold the corners into the center of the pan, doubling the crust. Egg wash the top of the puff pastry and place the pan into the oven. Bake approximately 25 to 30 minutes or until the puff pastry is a golden brown and fully cooked. Remove the tart from the oven and flip the tart out onto a large plate and allow it to cool. Cut into 6 pieces and serve alongside a green salad.