Preheat oven to 350º F. Grease 13x9" deep baking dish.
Arrange the croissants in an even layer in the baking dish and lay the butter over the croissants. Transfer dish to the oven and bake 5 to 8 minutes, until the croissants are lightly toasted.
Meanwhile, heat olive oil in a large skillet over medium heat. When the oil shimmers, add mushrooms and season with salt and pepper. Cook, undisturbed, for 5 minutes or until golden. Stir and continue cooking for 3 to 5 minutes until the mushrooms have caramelized. Add garlic and thyme, and cook an additional minute. Remove from the heat and set aside.
In a large bowl, whisk together heavy whipping cream, milk, eggs, mustard, cayenne pepper, and a pinch each of salt and pepper. Add the toasted croissants, mushrooms, prosciutto, and Brie cheese, and gently toss to combine. Pour the mixture back into the casserole dish. Top evenly with Swiss cheese and sprinkle with Everything Bagel spice.
Cover the baking dish with foil and bake for 30 minutes. Remove the foil and continue baking for 10 to 15 minutes or until the croissants are golden on top. Serve warm.
Notes
To get those "day old" croissants, just leave them out (uncovered) overnight. I forgot to do this the first time I made this though and the casserole was still just fine!
To make a day ahead, prepare the dish through Step 3. Combine the croissants and mushroom mixture and store in an airtight container in the fridge. When ready to assemble, combine the croissants/mushroom with the egg mixture in a large bowl. Continue recipe as directed!