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Line a large baking sheet with parchment paper or foil; set aside. Cut off and discard dark green ends of leeks. Halve white portions of leeks lengthwise. Cut into 1/4-inch-thick pieces (you should have about 4 cups).
In a large skillet, heat oil and butter over medium heat. Add leeks, salt, and cayenne pepper. Cook for 5 minutes, stirring occasionally. Reduce heat to medium-low. Cover and cook about 15 minutes or until leeks are tender but not brown, stirring occasionally. Remove from heat. Stir in thyme.
Preheat oven to 400°F. Unfold puff pastry. Cut off one-third of the pastry at the fold; set aside. On a lightly floured surface, roll the remaining pastry into a 14x7-inch rectangle. Place rectangle on the prepared baking sheet. Cut small leaf shapes from the reserved pastry.
In a small bowl, combine egg and water. Lightly brush pastry rectangle with egg mixture. Fold edges over to create a 3/4-inch ridge; lightly brush ridge with egg mixture. Gently press leaves onto ridge; lightly brush leaves with egg mixture. Sprinkle bottom of pastry with Parmesan cheese.
Bake for 12 minutes. Spread leek mixture over bottom of pastry (if puffed, pastry will sink when topped with the leek mixture). Bake about 5 minutes more or until edges are golden brown.
Immediately place Brie cheese on top of tart and sprinkle with olives. Let stand for 5 minutes before serving.