Heat oven to 400°. Line a baking sheet with parchment paper.Advertisement
Split the leeks lengthwise, slice crosswise into 1-inch chunks, and wash and dry well. Melt the butter in a medium skillet over medium-low heat. Add the leeks and shallot. Cook, stirring, until softened but not browned, about 7 minutes. Remove pan from heat.
Unfold both pieces of puff pastry. Brush 1/2 inch of the bottom edge of one sheet with a bit of beaten egg and place on parchment-lined baking sheet. Overlap the top 1/2 inch of the second sheet and press into 1 even sheet. Fold and press all edges inward to form 1 inch rim around pastry sheet. Brush rim and outside edges with beaten egg.
In a medium bowl, beat remaining egg with goat and blue cheeses. Spread cheese inside pastry shell. Top with leek mixture and sprinkle with thyme and some freshly ground black pepper.
Bake at 400° for 30 minutes. Sprinkle with more fresh thyme.
Use only the white and light-green parts of the leek. To clean, slice in half lengthwise, leaving the root end intact. Submerge in water several times to wash away all dirt.