For the strata: In a large saucepan over medium-high heat, melt the butter. Add the garlic and onion and cook until the onion is translucent, about 5 minutes. Add the oyster and shiitake mushrooms and chopped thyme and saute until the mushrooms are tender, about 5 minutes. Remove the mushrooms to a plate. In the same saucepan, add the sausage and cook until no longer pink, about 5 minutes. Transfer the sausage to the plate, leaving the drippings behind for gravy (refrigerate the drippings until ready to make the gravy).
In a large bowl, whisk together the milk, cream and eggs. Season with salt and pepper. Add the cheese and reserved mushrooms and sausage to the mixture. Mix in the bread cubes. Refrigerate for at least 3 hours and up to 24 hours.
Preheat the oven to 325 degrees F.
Grease a bundt pan with oil. Add a little flour to the pan so the strata doesn't stick. Add the strata mixture to the pan and bake until deep golden brown, 50 to 60 minutes. Let cool slightly, then transfer to a plate.
For the gravy: While the strata is baking, heat the reserved sausage drippings in a saucepan over medium heat. Add the butter and cook until melted. Add the flour and continue to cook until the mixture starts to thicken, about 5 minutes. Gradually whisk in the chicken broth while cooking. Add a pinch of salt and some pepper.
Serve the strata with the warm gravy and garnish with fresh thyme sprigs.