Preheat the oven to 350 degrees.
Heat the oil over medium-high heat in a heavy, wide skillet and add the mushrooms. Cook, stirring often, until the mushrooms begin to soften, add the shallot, thyme, garlic and salt and pepper to taste. Continue to cook for another minute or two, until the mushrooms are tender and fragrant. Stir in the kale, let the kale wilt and remove from the heat.
Spray a 2-quart baking dish. In a large bowl, combine the bread cubes, the mushrooms and kale, and cheese and toss together. Transfer to the prepared baking dish.
Beat together the eggs and milk. Add salt and pepper to taste, and pour over the bread mixture. Toss to combine and let sit for 10 minutes while the bread absorbs the liquid.
Transfer to the pre-heated oven and bake 45 minutes, until puffy and golden brown. Remove from the oven and serve hot or warm.