Let the broth cool down a bit to warm
Slice the onion in very fine pieces
in a flat iron, melt the butter and fry the onions, first at high, quite quickly at low The onions should not become brown. This may take 20 min
Then slice the mushrooms in narrow slices and add to the mixture. Heat on low
When vegetables are cooked, switch off the heat and sprinkle the flour over the vegetables and mix with a wooden spoon or a whisk, depending on the material of the pan.
Then put the heat on low, assure the flour is well mixed.
Optionally add the milk and stir well. Adding the milk will aid the binding power of the butter/flour mixture, but milk could be avoided and instead broth can be used as well.
After the mixture tends to warm through and becomes thicker, add in parts the browth, mix well and heat sufficiently to make a creamy ragout. Ensure the flour mixture has slowly cooked through for a few minutes
Add the finely sliced parsley leaves and mix
If too thick, add some more broth. If too thin, add some of a mixture of flour and cold broth and mix well, thereafter heating it again until the ragout puffs and cooks through
Taste the ragout, and use salt, pepper or some hydrolyzed wheat proteins to enrich the taste if so needed