Thinly slice the mushrooms then finely chop the slices.
Heat a medium saute pan over high heat. Add 2 tablespoons of the butter; when it starts to foam, add the mushrooms. Cook for 5 to 8 minutes, until the mushrooms begin to take on a golden brown color and all of the moisture has cooked off. (For the best flavor, it's essential to get a good sear and color on the mushrooms. If the mushrooms begin to stick to the pan a little, that's good as long as they don't turn black and burn. If the bottom of the pan starts to get too dark before you are ready for the next step, just add a little of the chicken broth or water and whatever is stuck will come right off the bottom of the pan.).
Add the onion, salt, pepper, thyme, and cook 5 minutes more, until the onion is soft. Add the chicken broth, bring to a boil, reduce heat to medium, and reduce the liquid by at least two-thirds over medium heat-there should only be about 3/4 cup of liquid left in the pan.
Transfer the liquid to a medium saucepan. Add the remaining 4 tablespoons of butter and gently stir over medium heat until melted. Whisk in the flour and the nutmeg. Stir for 2 minutes and add the cream. Cook over low heat for 7 minutes, then let cool.