Step 1: Brown the Butter: Place 1/2 cup of butter in a small saucepan over medium heat. Melt the butter, then continue cooking while stirring constantly. The butter will foam and then develop little brown specks. Once it smells nutty and looks golden brown, remove it from the heat immediately to prevent burning. Set aside to cool slightly.
Step 2: Prepare the Bread: Grease a 9×13 inch baking dish with butter or non-stick spray. Arrange the cubed brioche bread evenly in the dish. Try to spread it out so the custard can reach every piece.
Step 3: Whisk the Pumpkin Custard: In a large mixing bowl, whisk together the eggs, pumpkin puree, whole milk, heavy cream, brown sugar, maple syrup, vanilla extract, pumpkin pie spice, and salt. Slowly drizzle in the cooled brown butter while whisking constantly to ensure the eggs do not scramble.
Step 4: Soak the Bread: Pour the pumpkin mixture evenly over the bread cubes in the baking dish. Use a spatula or your hands to gently press the bread down, making sure every piece is coated in the custard. Cover the dish and refrigerate for at least 30 minutes, or overnight for the best texture.
Step 5: Make the Streusel Topping: In a medium bowl, combine the flour, brown sugar, and cinnamon. Use a fork or a pastry cutter to cut in the 4 tablespoons of cold butter until the mixture looks like coarse crumbs. Sprinkle this topping evenly over the soaked bread.
Step 6: Bake the Casserole: Preheat your oven to 350°F (175°C). Bake the casserole uncovered for 45 to 50 minutes. The top should be golden brown and crispy, and the center should be set but still slightly moist. Let it rest for 5-10 minutes before serving.