Drain 2 tablespoons of the oily liquid from the tuna cans into a skillet and heat it on medium. Drain the rest of the excess liquid out, but don't press the tuna so it stays moist.
Add the onion, carrot, and celery and saute for 5-7 minutes until softened.
Add the tuna, garlic powder, pepper, and salt, and saute for an additional 2-3 minutes. Remove from the heat and let cool for 5 minutes.
Transfer the tuna mixture to a mixing bowl. Add the mayonnaise, Dijon, cheddar, basil, and parsley and stir to mix. Set aside.
Preheat the oven to 400 F.
Unfold the puff pastry sheets and roll them out slightly on a lightly floured surface to enlarge.
Cut each sheet into 12 small squares.
Divide the tuna mixture evenly in the center of the squares.
Fold up 2 opposite corners of a pastry square to cover the filling and pinch the ends together with your fingertips.
Fold the 2 remaining corners in the same way and then pinch the 4 open seams together to seal. Fold and seal the remaining squares in the same way.
Place the pastries on a parchment paper-lined baking sheet and cut 4 slits in each pastry.
Beat the egg in a small bowl with the water.
Brush the beaten egg over the tops of the pastries.