Preheat oven to 400 degrees.

Melt the butter in a skillet over medium low heat and caramelize the onions until they are nicely browned and jammy. This should take about 20-25 minutes. I keep a cup of water by the stove and as the sugars brown and stick to the bottom of the pan, I add a couple tablespoons of water to help deglaze the pan and incorporate the sugar back into the onions. Season with salt and set aside.

On a lightly floured surface, roll the puff pastry into an even rectangle that measures approximately 10 x 14 inches. Score a 1/2 inch border around the edges using a sharp paring knife. Using a fork, prick the puff pastry within the border. Bake for 10 minutes until puffed and golden brown. Remove from oven and lightly press down the center of the puff pastry. Spread the caramelized onions in a very thin layer and top with the cheese. Trim the asparagus to fit within the frame of the puff pastry, crosswise. Toss the asparagus with olive oil and season with salt and pepper. Arrange the asparagus in a single layer, alternating the tips and ends. Bake in oven for 15 minutes, or until the asparagus is crisp tender and bright green. Pencil thin spears may be done in 10 minutes, while medium thick asparagus will take the full 15 minutes or more. Remove from the oven and when ready to serve, cut into squares with a sharp knife.