Sauté onions and mushrooms in a medium skillet until soft. Add the garlic and cook for another 1-2 minutes, then remove from the skillet and set aside.
Add vegetables back to the pan and bring to a boil. Turn the heat down a little and simmer the mixture (that means cook it just below boiling) until it is thickened. Remove from the heat and set aside.
Melt remaining ¼ cup of butter in a clean skillet. Add in the dill and crushed corn flakes. Lightly toast the corn flakes, then sprinkle the mixture onto the chicken breasts in the casserole dish.