Preheat the oven to 375ºF. Spray a large (3-quart) casserole dish with an oil mister and set aside.
Add the olive oil to a medium skillet set over medium heat. When hot, add the onion and cook, stirring occasionally, until it starts to soften, about 2 minutes. Add the garlic and cook, stirring, for 30 seconds, until fragrant. Add the apples and cook, stirring occasionally, until softened, about 5 minutes. Stir in the salt, pepper, sage and thyme. Pour into a large bowl and add the bread cubes and cheese.
In a small bowl, whisk together the pumpkin puree, broth and egg until combined. Pour it over the veggies, bread and cheese and toss until everything is combined and the pieces are evenly coated.
Transfer the mixture to the prepared baking dish and cover with aluminum foil. Bake for 30 minutes. Remove the aluminum foil and bake until the top is golden brown, about 20 more minutes more.