Mix breadcrumbs, parsley, cheese, remaining garlic, remaining 2 tablespoons of oil, and a light sprinkling of salt and pepper; set aside. Heat oven to 350 degrees. In a 13- by 9-inch or similar size pan, layer potatoes in 5 layers, seasoning the first 4 with a very light drizzle of olive oil, a light sprinkling of salt and pepper, and a portion of caramelized onions and thyme. Add the final layer of potatoes, pouring broth mixture over to nearly cover the potatoes. Sprinkle with breadcrumb mixture and bake until potatoes are cooked, and breadcrumbs are golden, about 1 hour. For a darker crumb topping, turn on the broiler and continue to cook to desired hue. Remove from oven, let rest a few minutes, then serve.