Made the pie dough. In a large bowl, whisk together 1 ¼ cups flour, 1 tbsp sugar, ¼ tsp salt and ⅛ tsp pepper. Rub butter in with your fingers or a pastry cutter until broken down into small pea-sized crumbs. Fold in ¼ cup parmesan until evenly distributed. Add cold water, a little at a time, stirring in with a fork as you go. Add just enough that the mixture starts to come together into a shaggy mass. Turn mixture out onto a clean work surface and knead gently with your hands just until mixture comes together. Shape into a disc, wrap tightly in plastic, and chill in the refrigerator for at least 30 minutes before rolling out. While dough is resting, prepare the leeks. Slice leek in half lengthwise and rinse thoroughly, then slice into thin half-moons. Melt butter in a saucepan over medium-low heat. Add sliced leeks and sprinkle with salt and pepper to taste. Cook, stirring frequently, until soft and melty, about 5-7 minutes. Set aside to cool completely.
Next, cook the mushrooms. Add 1 pound of chopped mushrooms to a saucepan over medium heat, stirring often. You may have to cook them in two batches so as not to overcrowd them. Let them release their water, then continue to cook until all the water has cooked off. Melt 1 tbsp butter into the pan, swirling to coat the mushrooms. Season with salt and pepper to taste. Set aside to cool completely.
Make the goat cheese spread. In a small bowl, beat together 4 oz soft goat cheese with 1 tbsp chopped fresh herbs, 1 tbsp honey, and 1 tsp lemon juice until well combined.
Assemble the galette. Roll dough out onto parchment paper into a large circle about ⅛-inch thick. Carefully move parchment to a baking sheet. Spread herbed goat cheese over center of dough, leaving a 2-inch border around the edges. Top with cooled prepared leeks, then with the prepared mushrooms. Fold up the edges, pleating as you go. Brush pie dough with beaten egg and sprinkle with parmesan, if desired. Chill in the refrigerator or freezer until cold and dough feels solid, at least 20 minutes.
Bake. While chilling, preheat oven to 375° F. Bake galette until crust is golden brown, about 35-40 minutes. Allow the tart to chill at least 15 minutes before slicing into it. Serve warm, room temperature, or cold.