Hard-boil the eggs, cut in half, mash the yolks and, separately, dice the whites.
Slice the onions finely and put into a pan with some butter; cook gently, with the lid on, so that the onions remain soft and transparent and do not get brown.
When they are just soft, and of a pale golden colour, sprinkle them with flour, stir in a little water or milk, and simmer for 30 minutes on a gentle heat.
When the resulting puree is cooked, mix into it half of the mashed cooked egg yolks and all the chopped whites; reheat for 1 minute, then pile the mixture on to pieces of toast.
Garnish with the remaining egg yolks and some chopped parsley.