Place the prosciutto slices on a dry paper towel. Lay another towel on top. Cook for 1 minute on high in the microwave. Remove top paper towel and allow the prosciutto to cool (meat will be slightly flexible but will crisp upon cooling). When completely cool, crumble prosciutto and set aside.
In a medium bowl, combine goat cheese, figs, crisped prosciutto, and tarragon. Stir until well-blended.
In a small bowl, beat egg and water with a fork and set aside. Lightly flour a work surface. Roll out the thawed puff pastry sheet to a 12x12-inch square. Using a pizza cutter, cut into 16 (3x3-inch) squares and then brush the edges of the squares with the egg mixture.
Place about 2 teaspoons of the goat cheese mixture in the center of each square. Fold the pastry over the filling to form triangles. Press the edges to seal. Crimp the edges with a fork.
Arrange 8 of the pastry triangles in the air fryer, leaving as much space as possible in between them. Set air fryer to 400°F for 10 minutes or until pastry is golden brown. Repeat with remaining pastry triangles.
If you don't own a pizza cutter yet, I highly recommend you get one for this recipe. It makes preparing the puff pastry so much easier and there's tons of other great applications for it.
Make sure you leave plenty of space between the pastry triangles when you air fryer them and don't try to cook them all in one batch! If they don't have enough room for the air to circulate around them they will stay doughy, and nobody wants that.
If you don't have an air fryer you can put them on a cookie sheet and bake them in the oven at 400F for 12-15 minutes.