Butter 13x9" glass baking dish. Cut bottom crust and shor ends off bread and discard. Cut remaining bread with crust into 1" cubes (about 10 cups loosely packed). Place cubes in very large bowl. Add oil, thyme, and garlic; toss to coat. Spread cubes out on large rimmed baking sheet. Sprinkle with salt and pepper. Bake until golden and slightly crunchy, stirring occasionally, about 20 minutes. Return toasted bread cubes to same very large bowl.
Melt butter in large skillet over medium-heat. Add mushrooms, onion, celery, and bell pepper. Saute until soft and juices have evaporated, about 15 minutes. Add sauteed vegetables and parsley to bread cubes.
Whisk heavy cream, eggs, salt, and ground pepper in large bowl. Mix custard into bread and vegtetables. Transfer stuffing to prepared dish. SPrinkle cheese over. Can be prepared 1 day ahead. Cover and refrigerate.
Preheat oven to 350 degrees. Bake stuffing uncovered until set and top is golden, about 1 hour. Let stand 15 minutes.