Combine chicken broth and water in a large saucepan over high heat.
Slice the white onion in half, then coarsely chop one half (save the other half for later).
Add coarsely chopped onion, carrot, celery and salt to the saucepan and bring to a boil. Reduce heat and simmer for 10 minutes, or until the onions start to become translucent. As the broth simmers, heat up 1 cup of vegetable oil in a small saucepan over medium heat.
Slice the remaining white onion into very thin slices. Separate the slices, dip the slices into the milk, then into the flour. Fry the breaded onions, a handful at a time in the oil until golden brown. Drain on a paper towel.
When the soup has simmered for 10 minutes, strain the vegetables out of the broth and toss them out. Pour the broth back into the pan and keep it hot over low heat.
To serve soup, ladle about 1 cup of broth into a bowl.
Drop a few pieces of fried onion into the soup, followed by 6 to 8 mushroom slices (about 1 mushroom each) and a couple pinches of diced green onion. When the fried onions sink to the bottom of the bowl (a couple of minutes), serve the soup.