In a medium sauce pan over medium heat, melt the butter. Sautee the chopped mushrooms in the melted butter until they release their liquid and shrink in size, about 5 minutes.
Sprinkle the flour over the cooked chopped mushrooms and stir to coat the mushrooms in it.
Slowly stir 1 cup of the warmed milk into the mushroom flour mixture. You want the milk to become absorbed by the flour on the mushrooms and form a sort of thick sauce.
Repeat this process until you have no more milk left, and then stir in the chicken broth as well. Simmer for 3 to 5 minutes, or until the mixture begins to thicken.
Stir in the green beans, soy sauce, and the remaining salt and pepper. The sauce should coat the green beans, but the pan should not be dry. It should be like a thick green bean soup. If it's too thin, crank the heat up the medium high and simmer the mixture until it thickens. If it's too thick, add either more milk, chicken broth, or water ( whatever you have on hand ) ½ cup at a time until it thins.
Pour the green beans into a 2 ½ - 3 quart baking dish.
Bake for 20 - 30 minutes, or until the edges of the dish begin to brown.