Preheat the oven to 350°. In a large rondeau pot add 3 tablespoons of olive oil over high heat and add the mushrooms to caramelize or well browned, stirring every 2 minutes or so.
Next, turn the heat down to medium and scrap the mushrooms to one side of the pan add 1 tablespoon of butter, shallots, garlic, and celery to the other side of the pan and sauté until lightly browned.
Stir in the flour completely with the vegetables to create a roux and then pour in the milk and turn to high heat to activate the roux to help thicken up the sauce.
Once the sauce is thickened, which will almost immediately happen once it begins to boil, finish it with thyme, salt, and pepper and mix until combined. Turn the heat off and set aside.