Slice off top and bottom of onions. Chop both onions. Coat the bottom of a medium size sauté pan with olive oil. Heat pan over medium-high heat. When oil is hot but not smoking add onions. Stir to coat the onions. After 10 minutes add a pinch of salt. Continue to cook onions stirring every few minutes. Do not let the onions get too brown turn heat down if necessary. Onions should take about an hour to cook. When the onions are a dark brown add a dash of balsamic vinegar and cook for another minute or two. Take off of heat and let cool.
In a large mixing bowl mix together ground chuck, ketchup, yellow mustard, salt and pepper. Form into 16 equal sized patties. Heat a large skillet over medium-high heat and cook patties for 1-2 minutes per side. When all patties are cooked set aside.
Preheat oven to 350°. Line a rimmed baking sheet with parchment paper. On a lightly floured surface roll out one sheet of puff pastry. Cut pastry in 4 equal strips lengthwise then in half so that you have 8 pieces. Place one patty on each piece and top with a little of the caramelized onions and then a little of the cheese. Brush the edges of the pastry with the egg wash. Wrap the pastry up and around the burgers and pinch edges together to seal. Continue until you have all 16 burgers wrapped. Brush the tops of all of the burgers with the egg wash.
Place burgers on prepared baking sheet and bake for 12 to 15 minutes or until puffed and lightly browned on top. Let stand for 5 minutes before serving.