Preheat the oven to 350 degrees F. Butter the bottom and sides of a 13-by-9-inch baking dish with 1 tablespoon butter.
Whisk together the pumpkin puree, half-and-half, granulated sugar, eggs, 2 teaspoons pumpkin pie spice, 1/4 teaspoon cardamom and 1/2 teaspoon salt in a large bowl until well blended.
In another large bowl, combine the oats, brown sugar, flour, remaining 1/2 teaspoon pumpkin pie spice, 1/4 teaspoon cardamom and 1/2 teaspoon salt, breaking up any lumps of brown sugar with a fork. Melt the remaining 7 tablespoons butter and stir into the flour mixture until it is completely moistened and crumbly.
Spread the pumpkin mixture evenly in the bottom of the baking dish. Scatter the oat mixture on top. Bake until the pumpkin is set around the edges but still jiggly when shaken, and the topping is brown and crisp, about 50 minutes.
Serve room temperature or slightly warm with whipped cream on the side.
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