Over medium-high heat, heat a Dutch oven or heavy soup pot. Add olive oil and mushrooms.
Stirring occasionally, cook mushrooms until they begin to release their juices (about 6-7 mins). Season with salt and pepper to taste.
To the pan, add white parts of the onion. Cook 1-2 minutes, making sure to stir once or twice.
Increase the heat to high and add Marsala wine. Cook until wine is reduced to about ⅓ of its original volume (about 4-5 minutes).
To the pan, add chicken stock and shredded chicken. Bring to a boil, and then reduce heat to just slightly below medium. Simmer until thoroughly heated (about 4-5 minutes).
Remove from heat and serve. Garnish with green parts of onion.