Preheat the oven to 375 degrees F (190 degrees C). Get out 2 muffin tins that each have 12 wells and spray them both well with cooking spray.
Take each sheet of Puff Pastry (2 sheet) and cut it into 12 squares. Do this by cutting along the fold seams to cut the sheet into thirds, then cut each third into 4 equal squares. Press each square into a muffin well to fill both tins.
In a food processor, combine Smoked Gouda (4 ounce), Cremini Mushroom (16 ounce), Garlic (2 clove), Salt (1 pinch), Ground Black Pepper (1 pinch), Dried Tarragon (1/2 teaspoon), and Dried Basil (1/2 teaspoon). Run the processor until you have a fine mixture that still has a little bit of texture.
Use a big, heaping tablespoon to scoop the filling into the center of each puff pastry square. Bring the corners together on top of the filling to help enclose it, but it's okay if some goes through the sides.
Brush the Egg (1) on each puff, then bake them for 25-30 minutes. The puff pastry should be golden and crisp, with the filling set.
Let puffs cool for a few minutes, then run a knife gently around the edge of each one to help loosen them from the tin. Serve immediately and enjoy!