Preheat your oven to 375 F.
Cut the mushrooms into large bite size pieces.
In a sauce pan, heat the butter and oil over medium high heat. Add the mushrooms with a good pinch of salt and pepper and cook until the mushrooms are starting to get soft and are almost cooked, about 10 minutes. Add the shallots and sauté for another 4 or 5 minutes. Add the stock, bring to a simmer and reduce the heat. Cook for about 10 more minutes, ideally the liquid should be reduced to about 1 cup or so. The mushrooms will be "wet" but not swimming in liquid. Adjust seasoning with salt and pepper as needed.
Brine the halibut for 15 minutes (not more!) in the brine solution. Rinse and pat dry. Alternatively, season well with salt and let sit for 20 to 30 minutes before cooking.
Transfer the mushrooms and their liquid to a large deep baking dish. Set the halibut fillets on top and place a thyme sprig or two over each fillet. Cover the baking dish and bake in the oven for about 11 to 15 minutes. The fish should be flaky and hot in the center. Ideally use a thermometer to ensure the fish is about 127F to 130F in the center.
Serve right away in deep bowl with the mushrooms on the bottom and the fish on top with plenty of the mushroom stock.
