Place chicken, 1 egg white, salt and pepper in processor. Using on/off turns, process until smooth. With machine running, gradually add 1/2 cup cream. Set mousse aside.
Melt butter in large skiller over medium-high heat. Add chopped fresh mushrooms; saute until soft, about 4 minutes. Cool to lukewarm. Stir mushrooms and tarragon into mousse.
Arrange 18 wonton wrappers on work surface. Lightly whisk remaining egg white. Place 1 tbsp mousse in center of 9 wontons. Place remaining 9 wonton wrappers atop mousse, matching corners and pressing to remove air pockets, and seal edges tightly. Transfer to baking sheet. Repeat with remaining wrappers, egg white, and mousse.
Can be made 4 hours ahead. If so, cover and chill.
Place dried morels in medium metal bowl. Pour 1 cup boiling water over. Cover; let stand at room temperature 1 hour. Boil fava beans in medium saucepan until just tender, about 4 minutes. Drain; set aside.
Using slotted spoon, transfer morels to medium saucepan. Add mushroom soaking liquid to pan, discarding any sediment at bottom of bowl. Stir in Madeira. Boil until almost all liquid evaporates, about 7 minutes. Add broth and remaining 1/2 cup cream. Boil until slightly thickened, about 3 minutes. Stir in fava beans.
Sauce can be made 2 hours ahead. Let stand at room temperature. Bring to simmer before using.
Working in batches, cook ravioli in large pot of boiling salted water until ravioli float to surface, about 3 minutes. Drain. Divide ravioli among 6 plates. Spoon sauce around ravioli and serve.
PLEASE NOTE: Cooking time is approximate. No times were given, so it is just a guess. If you make this, please let me know how long it takes!