In a small bowl, beat together the eggs and water until blended and set aside. Place the cookies in a heavy lock-top plastic bag and, using a rolling pin, crush the cookies to make fine crumbs. (Alternatively, crush the cookies in a food processor.) Set aside. Combine the granulated sugar and cinnamon in a small bowl and set aside.

Have ready 2 rimmed baking sheets, preferably nonstick. For pans without a nonstick finish, line the pans with parchment paper or nonstick baking liners.

Unfold the pastry sheets and lay them flat on a large cutting board. Using a sharp knife, cut each pastry sheet into quarters, creating eight 5-inch squares. Place 4 squares on each prepared baking sheet, spacing them about 2 inches apart. Fold in the 4 sides of each pastry square about half an inch toward the center, so the edges will be a double thickness. Use your little finger to make indentations about half an inch apart around the edge of each pastry, creating a fluted effect. Place the pastry in the refrigerator for 15 minutes. (The pastry can be prepared up to this point, covered lightly with plastic wrap, and refrigerated overnight.)

Position one rack in the center of the oven and another in the lower third. Preheat the oven to 400 degrees. Place the apricot jam in a small bowl and microwave just until melted, about 10 seconds. Alternatively, heat in a small saucepan over low heat until melted. Set aside. Prepare the pears.

Remove the pastry from the refrigerator. Using a pastry brush, brush the pastry with the beaten egg mixture. Place a rounded tablespoonful of the cookie crumbs in the center of each pastry shell and use your finger to spread it around evenly, keeping the crumbs off the edges. Lay the pear slices in a tight overlapping pattern over the cookie crumbs. Sprinkle the pears with the cinnamon-sugar mixture. Lightly brush the pears with the melted preserves.

Bake the tarts for 15 minutes, switch the position of the baking sheets and rotate the sheets 180 degrees, and continue to bake until the crusts are beautifully browned, 30 to 35 minutes longer. Let cool slightly on the pans on wire racks. (The tarts can be baked up to 8 hours ahead, allowed to cool completely, and then reheated in a 400 degree oven for 10 minutes just before serving.)

Using a small fine-mesh sieve, dust the top of each tart with the confectioners’ sugar. Serve warm on individual plates with a scoop of ice cream on the side.