Preheat the oven to 400°F. Line a baking sheet with parchment paper.

Cut the pastry sheet in half, lengthwise down the middle so that you have two equally sized pieces. Onto the prepared baking sheet, arrange both puff pastry pieces at least 2 inches apart. Fold over the edges about 1/2-inch on each side to form a border all around.

Melt 1 tablespoon of butter and use a pastry brush to lightly brush the surface of the pastry dough, including the folded over edges.

On the stovetop, set a large pot of water over medium-high heat. Bring to a boil then add the green beans. Cook the green beans for 2 minutes, until just crisp-tender and bright green in color. Drain in a colander and rinse with cold water to stop the cooking process.

Place a layer of paper towels or kitchen towels onto the countertop. Spread the cooked green beans onto the towels in an even layer to dry completely.

To a large skillet over medium-high heat, add 2 tablespoons of butter. When the butter is melted, add the mushrooms and thyme. Saute for 3 minutes, until softened and fragrant. Season with salt and pepper.

Add the garlic and continue cooking for an additional 30 seconds. Remove the mixture from the skillet and set it aside. Keep the heat on and return the skillet to the stovetop.

Into the skillet over medium-high heat, add 1 tablespoon of butter. When the butter is melted, add the sliced shallot. Cook for 5 minutes, stirring frequently until the shallots are lightly crispy and golden in color. Remove from heat.

Divide the green beans in half and arrange an equal amount onto each pastry sheet. Follow with the mushrooms and then the shallots. Sprinkle evenly with feta cheese.

Bake for 15 minutes, or until the puff pastry is golden in color then serve warm or room temperature.