1. Bring stock and 1/2 cup water to a simmer in a small saucepan. Add the mushrooms. Let stand until soft, about 20 minutes. Remove mushrooms with a slotted spoon, and transfer to a work surface; coarsely chop. Set aside. Pour soaking liquid through a fine sieve into a medium bowl; discard solids. Return soaking liquid to pan; set aside.

2. Separate the leaves from the stems of chard; reserve stems. Coarsely chop leaves. (You should have about 4 cups of leaves.) Set aside. Cut stems into 1/4-inch dice. Add stems and thyme to soaking liquid in pan, and bring to a simmer. Remove from heat; let stand 10 minutes. Pour through a fine sieve into a medium bowl. Reserve 1/4 cup simmered chard stems for garnish; discard remaining solids. Cover broth, and set aside.

3. Heat oil in a large skillet over medium heat. Add prosciutto; cook, stirring occasionally, until crisp, about 5 minutes. Transfer to a plate. Add onion to the skillet; cook, stirring occasionally, until translucent, about 3 minutes. Stir in mushrooms and chard leaves. Cook, stirring occasionally, until chard is wilted, about 5 minutes. Transfer to a medium bowl; let cool completely. Coarsely chop prosciutto; stir into chard mixture. Add Parmesan and stir to combine.

4. Working with a few wrappers at a time, put 1 heaping tablespoon mixture onto the center of each wrapper. Brush edges with water. Top each filled wrapper with another wrapper. Press edges and centers to seal. (Uncooked ravioli can be frozen in a single layer on a baking sheet and then stored in a resealable plastic bag up to 1 month.)

5. Bring a large pot of water to a boil. Add salt. Add ravioli in batches; cook until they rise to the top and are heated through, about 3 minutes. Carefully transfer 3 ravioli to each serving bowl; cover. Repeat with remaining ravioli.

6. Bring the broth to a simmer. Spoon 1/4 cup broth into each bowl; garnish with reserved chard stems.