For the casserole: Generously grease a 3-quart casserole dish with the butter. Shingle the bread slices, in rows, over the bottom of the dish so that they are overlapping slightly (like rows of slanted books).
Whisk the eggs, sugar, pumpkin pie spice, vanilla and salt in large bowl until smooth and well combined. Add the milk and cream to the bowl and whisk until well combined. Pour the egg mixture evenly over the bread slices, making sure that all slices are well coated. Cover the casserole dish with plastic wrap and refrigerate for at least 8 hours or up to overnight.
Position an oven rack in the center of the oven, and preheat to 350 degrees F.
For the brown sugar crumble: Whisk together the brown sugar, flour and salt in a medium bowl. Add the butter and work it into the dry ingredients with your fingers (or a pastry cutter) until the mixture is crumbly with pea-sized bits of butter. Mix in the pecans.
To assemble: Give the bread slices in the casserole dish a gentle push into the custard, dollop the pumpkin pie puree over the bread slices and sprinkle the crumble evenly over the top.
Bake the casserole until the top is puffed and browned and a knife inserted in the center comes out clean, about 45 minutes (it will still be a little jiggly when the casserole dish is tapped but it will continue to cook a bit out of the oven). Check the casserole about halfway through the baking time and tent it with foil if the topping is browning too quickly. Let rest 10 minutes before serving.
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