Heat the olive oil and the butter together in a large skillet over medium heat.
Add the mushrooms, thyme, shallots, garlic, salt, and pepper. Sauté for five to seven minutes, or until wilted.
Remove from the heat and allow to cool. Chop coarsely once cooled.
Preheat the oven to 325 degrees F.
Butter six custard dishes and set them inside of a large baking dish.
Use a large bowl to beat the eggs lightly. Add the cream and the milk and continue beating.
Add the mushroom mixture and beat again briefly to incorporate.
Pour the egg mixture into the buttered custard dishes.
Bring the water for the bain marie to a light simmer on top of the stove. Pour the hot water into the baking pan until it comes about half way up the outsides of the custard dishes. This will help to protect the custard from direct heat inside the oven.
Bake the custards for about 25 to 30 minutes, or until they're "set" around the edges but still jiggly in the center. The cooking time will depend on your individual oven and on the size of the custard dishes you're using, so watch them carefully.
Immediately remove the custard dishes from the baking pan as soon as you remove them from the oven.
Cool the dishes on a wire rack until room temperature. Cover with plastic wrap and refrigerate for about two hours before serving. Can be made up to one day ahead of time.
Dress the arugula leaves with the hazelnut vinaigrette dressing.
Unmold the mushroom timbales by running a knife around the outside rim.
Turn them upside down on individual serving plates, giving the mold a quick shake.
Garnish with the dressed arugula.