Heat the butter or oil in a skillet on medium high heat. Add the onions, seasoning with salt and pepper to taste. Cook until the onions have softened, about 5 minutes. Add the mushrooms, season with salt and pepper and cook for another 5-7 minutes, until the mushrooms are tender and golden. At first the mushrooms will exude a lot of liquid, but then eventually absorb it and begin to brown.
Add the cooked chicken, the onions and mushrooms to a food processor and pulse a few times until it's all evenly mixed in very small pieces. You don't want it to be a puree, but the chopped up chicken and mushrooms should be pretty tiny. Transfer to a large bowl.
Finely grate the hard boiled eggs on the small holes of a grater. (If you have a meat grinder, you can put the chicken, mushrooms and the eggs through the grinder instead of using the food processor.)
Season with salt and ground black pepper, if needed. Add the grated cheese and the chicken broth. You can add more or less broth, depending on how dry/wet the chicken and mushroom mixture is. The broth will help to keep the mixture together and later on when it is cooked, will help keep the filling from falling apart and from being too dry.