Cut stems into matchsticksizestrips. Cover and refrigerate stems;reserve for sauce.
Cook chard leaves in large pot ofboiling salted water until tender, about3 minutes.
Drain; cool. Squeeze chardleaves until very dry; place in processor.Using on/off turns, finely chop chard.
Addricotta, 1/2 cup Parmesan, shallot, egg,coarse salt, pepper, and nutmeg; processto blend.
Transfer mixture to medium bowl.
Add 1/2 cup flour; stir to blend. Cover andrefrigerate dough overnight.
Bring chicken broth, slicedshallots, and mushroom stems to boil inlarge saucepan. Reduce heat to mediumand simmer until mixture is reduced to 3cups, 35 to 40 minutes. Strain. Return brothto saucepan; discard solids in strainer.DO AHEAD: Can be made 1 day ahead. Coverand chill. Rewarm before continuing.Melt 2 tablespoons butter with 2tablespoons oil in heavy large skillet overmedium-high heat; add sliced mushroomcaps.
Sprinkle with salt and pepper; sautéuntil mushrooms are tender and browned,8 to 10 minutes.
Transfer to bowl.
Heatremaining 1 tablespoon oil in same skilletover medium heat.
Add reserved thinlysliced chard stems and sauté until tender,6 to 8 minutes. do ahead Mushrooms andchard stems can be prepared 2 hours ahead.
Let stand at room temperature.
Line rimmed baking sheet withparchment paper. Spoon some flour ontolarge plate. Working in batches, dropheaping teaspoonfuls gnudi dough (size ofsmall walnuts) onto plate with flour to formabout 36 gnudi. Using floured hands, gentlyshape each into 1 1/2-inch-long, 1/2-inch-thickoval. Tap off excess flour; transfer gnudi toprepared baking sheet. do ahead Can bemade 2 hours ahead. Cover and chill.Bring large wide pot of salted water toboil. Slide gnudi into pot; cook until tender,stirring occasionally, about 10 minutes.Meanwhile, whisk remaining 4tablespoons butter into hot broth. Seasonbroth to taste with salt and pepper. Coverto keep warm.
Divide mushrooms and chard stemsamong 6 bowls.
Add broth, dividing equally(about 1/2 cup each). Using slotted spoon,divide gnudi among bowls.