1. Cut away any tough stems and hard parts from the mushrooms, reserving the stems and hard parts. (If they do not have tough stems or hard parts, cut off about ¾ cup of stems for the broth.) Wash the stems and hard parts well and place in a food processor. Add 3 teaspoons of the thyme, 3 teaspoons of the rosemary, and the garlic. Process until finely chopped.
2. Transfer the stem mixture to a small saucepan and add 2 cups of water, the soy sauce, and vinegar. Bring to a boil. Scrape down the sides of the pan and reduce the heat to a simmer. Cover and cook over very low heat for 15 minutes.
3. Cut the mushrooms into chunks or slice longer mushrooms lengthwise and set aside. Cut off and discard about a third of the bottom of the asparagus stalks. Cut off the asparagus tips, then cut the remaining stalk diagonally into ¾-inch lengths. Set aside.
4. Strain the mushroom stem liquid into a bowl, pressing down on the mushroom mixture to extract as much liquid as possible. Discard the solids. Cover to keep warm. (You may prepare the pot pie to this point up to 2 days in advance. Reheat the mushroom broth to a simmer before continuing.)
5. Make a light brown roux by melting the butter in a heavy, medium-size saucepan over medium heat until foaming. Stir in the flour and whisk until smooth. Reduce the heat to low and cook, stirring frequently, until the roux is tan, without letting it stick or burn, about 4 minutes. Slowly pour the mushroom broth into the roux, whisking constantly to prevent lumps. When all the broth is added, increase the heat to medium and continue whisking, paying special attention to the corners, until the mixture comes to a boil. Reduce the heat and simmer, uncovered, for 10 minutes, stirring several times.
6. Add the mushrooms to the sauce and stir to coat. The mixture will look dry, but the mushrooms will give up a lot of liquid as they cook Simmer, stirring occasionally without letting the sauce boil or stick, for 10 minutes. Stir in the cream, chopped parsley, and remaining thyme and rosemary. -Simmer for 3 minutes. Remove from the heat and stir in the asparagus. Cool to room temperature. (The filling can be made to this point up to 1 day in advance. Cover and store in the refrigerator. Before continuing, warm the mushrooms gently over very low heat, adding water a few drops at a time, if necessary, to restore the filling to its original consistency. Heat just to room temperature, without letting the mixture simmer or boil.)
7. Divide the filling among six 5- to 6-ounce ramekins or custard dishes about 3½ × 2 inches.
8. Preheat the oven to 400°F.
9. To prepare the pastry, roll out the puff pastry on a lightly floured work surface to 1/8 inch thick Add flour as necessary to prevent sticking during the rolling. Measure the diameter of the ramekins and cut 6 circles ½ inch wider than the diameter. Brush the edges with the beaten egg and cover the dishes with the pastry, egg side down. Press the pastry firmly against the sides of the dish. The pastry should be taut across the top of the dish and firmly attached to the dish all the way around. Brush the pastry tops with the remaining beaten egg. Arrange the potpies on a baking sheet and refrigerate for 5 to 10 minutes.
10. Bake until the pastry rises into a dome shape and is well browned, about 20 minutes. Serve at once with a spoon and knife.