Preheat oven to 425° F / 220° C. Line a baking sheet with parchment and set aside.

In a large frying pan, add olive oil (or butter) and chopped leeks. Caramelize over medium-low heat until soft and golden brown, about 20 minutes. Season with salt and pepper, then transfer to a bowl and set aside to cool.

Once the leeks have cooled slightly, add crumbled cheese and chopped walnuts, and stir gently to combine.

Unroll puff pastry sheets. (If using homemade pastry, roll to about 1/4-inch / 6-millimeter thickness.) With a sharp knife, cut the pastry into 18 4-inch / 10-centimeter squares.

Transfer nine of the pastry squares to prepared tray. Add a spoonful of the leek mixture to the center of each square, allowing a 1/2-inch / 1-centimeter border around the edge.

With a pastry brush, brush border with egg wash, then top each with a second square of puff pastry. Use a fork to crimp the edges of each pastry.

Brush tops with egg wash, and use a knife to cut a small slit into the the top of each pastry.

Bake in preheated oven for 12-15 minutes, or until the puff pastry is flaky and golden. Remove from oven and sprinkle each pie with a bit of fresh thyme.

Serve warm or at room temperature.

Puff pastry hand pies are best served on the day they are made. Refrigerate leftovers in an airtight container and gently rewarm in the oven if enjoying leftovers the next day.