First, thaw the puff pastry according to package instructions. Most frozen puff pastries take a few hours to thaw in the fridge, so this will require a little advance planning.
Preheat the oven to 400℉/204℃. Line a baking sheet with parchment paper and sprinkle flour on it. Set aside.
Wash the vegetables and pat dry. Prepare and gather the rest of the ingredients so that you can work quickly when it's time to add the galette filling.
Warm olive oil in a large skillet over medium heat. Add the sliced leeks and mushrooms and salt and pepper. Cook, stirring often until the mushrooms have released their liquid and it's mostly evaporated. The mushrooms and leeks will be lightly browned. Pour in the balsamic vinegar to deglaze the pan, using your spoon to scrape any browned bits off the bottom of the pan. Remove from heat and stir in the thyme leaves.
When you have all of the ingredients prepared, take the puff pastry out of the refrigerator. If you are using the recommended Dufour brand, you don't need to roll it out. You can just put it on top of the floured parchment paper on the baking sheet and unfold it. Use a pizza cutter or knife to cut it into a 10- to 12-inch oval. If you really want it to be a true circle, you can roll the dough out and cut it into a circle. I just like the idea of using puff pastry for a true shortcut so I don't bother rolling it. You can even leave all of the dough and not cut it into an oval if preferred.If you are using homemade galette dough or another brand of puff pastry, you may have to roll the dough to get it to be about ¼-inch thick before cutting into a circle. Use a fork to prick the dough several times, leaving a 1-inch border all around the circle that's untouched.
Use your hands to crumble the goat cheese on the puff pastry within the border. Spread it out as best you can.
Spoon the mushroom leek mixture on top of the goat cheese.
Add the shredded gruyere cheese on top.
Fold in the edges of the puff pastry circle, pinching the dough as needed to hold it together.
Prepare an egg wash by whisking the egg vigorously with 1 teaspoon cold water. Brush this on the border of the circle. This will help the crust develop a nice glaze in the oven, but you can skip this step to save time if you want.
Bake the galette in the preheated oven for 20 to 23 minutes until the cheese is melted and the crust is golden brown. Let the galette cool for about 10 minutes. The pastry will puff up during baking but it quickly deflates while cooling.
Cut into slices, and enjoy!
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