Step 1
Preheat oven to 400 degrees F (200 degrees C).
Step 2
Place 1 piece of puff pastry on a baking sheet lined with a silicone mat or parchment paper. Lightly moisten one of the short edges with water. Connect the end of the other puff pastry to the moistened edge. Press pieces together to form a long rectangle trimming off loose ends with a dough scraper as needed.
Step 3
Cut the third puff pastry piece lengthwise into 1-inch strips. Moisten the outer edges of the long rectangle with your finger; stick the pastry strips on top. Poke shallow holes all over the bottom of the dough using a fork. Do not prick the raised border.
Step 4
Bake pastry shell in the preheated oven until it starts to turn golden brown about 15 minutes. Remove from oven. Reduce oven temperature to 375 degrees F (190 degrees C). Readjust the borders and push the bottom down with the flat side of a fork. Let cool completely while preparing the filling.
Step 5
Place butter and 1/3 cup plus 1 tablespoon sugar in a bowl; smear together using a spatula until combined. Whisk in egg until mixture is creamy. Add almond flour salt vanilla extract and almond extract. Mix thoroughly. Cut each pluot into 6 wedges. Fill pastry shell with the almond paste. Stick the pluots snugly but not too deeply into the filling.
Step 6
Bake in the preheated oven until browned and almond filling is set 30 to 40 minutes. Transfer tart onto a wire rack and let cool completely at least 40 minutes.
Step 7
Combine apricot jam and water in a microwave-safe bowl. Bring to a boil in the microwave. Let cool until warm. Brush glaze over the cooled tart.