Place bread cubes in a bowl. Cover and let stand until stale, 8 hours to overnight.
Preheat oven to 350 degrees F (175 degrees C). Grease a 7x11-inch baking dish with cooking spray.
Whisk pumpkin puree, eggs, milk, 1/2 cup melted butter, white sugar, 1/2 cup brown sugar, maple syrup, pumpkin pie spice, vanilla extract, and salt together in a large bowl. Add bread cubes a few at a time, folding until mixture is moist but not soupy. Pour into the baking dish.
Combine flour, 3/4 cup brown sugar, 1/2 cup butter, and cinnamon in a bowl; crumble together with your fingers. Stir in walnuts; sprinkle over the baking dish.
Bake in the preheated oven until center is set and crumb topping is golden brown, about 1 hour.