First, in a small bowl, mix together flour, salt, pepper, and paprika.

In another small bowl, mix together egg and milk.

In a frying pan, heat lard or oil over medium heat, about 1/2" deep.  Dip the separated, sliced onions into the milk/egg mixture and then the flour mixture.  Fry over medium heat until golden brown.  Put them on a plate lined with a towel and set aside.

Put the green beans in a steamer basket in a 4-qt. pot with some water on the bottom.  Cover it and turn it on low.  Steam the green beans for 10 - 15 minutes or until tender (don't let them lose color).

In a saucepan, melt the butter.  Add the chopped onions and mushrooms and saute until the onions are soft and the mushrooms are done.

Add the arrowroot powder and some salt and pepper.  Stir until smooth.  Add the stock and milk and cook and stir until thickened.

Put the steamed green beans into an 11x7 or 8x8 casserole dish.  Add the mushroom sauce and some of the fried onions.  Stir to combine, and add salt and pepper to taste.  Top with more fried onions.

Bake at 350 for 20 - 25 minutes, until bubbling.  Serve immediately.