Preheat the oven to 425 degrees F. Line a cookie sheet with parchment paper and set aside.
Blanch the green beans in boiling water for 1-2 minutes. Drain and place in cold water to cool. Then set on a paper towel to dry.
Melt 2 tablespoons of butter in a skillet over medium heat. Add the mushrooms and the thyme and saute for 3-5 minutes. Then add the garlic, and salt and pepper to taste. Saute another 2-3 minutes and remove from skillet and pull out the thyme sprigs.
Add the remaining butter to the skillet with the sliced shallots. Saute until golden-brown and crispy, 5-8 minutes. Meanwhile, cut the puff pastry down the middle and place both pieces 3-4 inches apart on the cookie sheet. Whisk the egg with a tablespoon of water, and brush the pastry sheets with egg. Neatly layer green beans down the length of each puff pastry rectangle.
On both puff pastry sheets, top the green beans with mushrooms, crumbled blue cheese, and crispy onions. Salt and Pepper lightly.
Bake for 15-20 minutes until the edges are golden. Serve warm or room temperature.