For the dressing
Combine all of the ingredients in a jar or leakproof container and shake well to combine. Set aside.
For the mushrooms
Marinate the mushrooms in the canola oil, olive oil, 1 tablespoon of the salt, thyme, and shallots for 1 hour. Preheat the grill over high heat. Grill until caramelized and crispy on all sides. Toss the mushrooms with the dressing, parsley, cheese, remaining salt, and freshly ground black pepper, to taste.