Roll one sheet of puff pastry to 1/4 inch thickness. Score a half-inch border around puff pastry. Spread half fig jam on top inside scored border. Spread frangipane on top of fig jam. Arrange fig halves on top of jam. Beat remaining egg with one tablespoon water and brush on top of tart, including the pastry edges. Repeat with second piece of puff pastry.
Frangipane adapted from Food and Wine