In an 11- to 12-inch frying pan over medium-high heat, stir onions and butter often until onions are limp and lightly browned, about 10 minutes.
Meanwhile, shell eggs and chop.
Mix a little broth with flour to make a smooth paste, then add to remaining broth and mix into onion mixture. Add milk and stir over high heat until boiling, then stir about 2 minutes more.
Add eggs to sauce and mix well.
Spoon sauce onto toast. Sprinkle with parsley and add salt to taste.