Preheat the oven to 350°F (176°C) with a rack in the center position. Spray a 6-well doughnut pan with baking spray and set aside.
Prepare Cardamom Crumble: Cut the cold butter into small cubes and place in a medium bowl. Add the remainder of the ingredients and work them into the butter with your fingertips until the mixture comes together in large clumps and crumbs. If the butter gets too soft or warm, place in the freezer for several minutes before continuing to mix. Once you have finished making hte crumble, place in the freezer until ready to use.
Bake Doughnuts: In a medium bowl, whisk together the flour, salt, baking soda, baking powder, ground cinnamon, and ground allspice. Set aside.
Place the softened butter and granulated sugar in a stand mixer, fitted with paddle attachment. Beat on medium speed for 2 to 3 minutes, scraping down the bowl with a spatula as necessary. Reduce the speed to low and add the egg, mix until fully incorporated. Scrape the bowl to ensure that all of the ingredients are incorporated evenly.
Add the pumpkin puree and vanilla, and mix over low speed until just blended (the mixture will look slightly broken). Add the flour mixture in three parts, alternating with two additions of milk. Beat at low speed until just combined–avoid over-mixing. Scrape the bottom of the mixing bowl to ensure that all of the ingredients are incorporated.
Transfer the batter to a piping bag fitted with a large round piping tip (alternatively, you can use a spoon to fill the doughnut pan, but it will be more difficult) and distribute the batter evenly in the greased pan. Remove the crumble from the freezer and distribute it onto the batter, pressing it slightly into the batter. The crumb topping will be generous, but you shouldn’t have any leftovers.
Bake the doughnuts for 16 to 18 minutes, or until the doughnuts spring back when lightly pressed. Allow the doughnuts to cool in the pan on a rack for 5 to 10 minutes, before removing (use a knife to release the edges from the donuts from the side of the pan). Remove and allow them to cool completely on a rack.
Dust with powdered sugar before serving (optional). Doughnuts best served the day of baking, but can be stored in an airtight container at room temperature for 2 to 3 days.
Tips for Success:
Ground cardamom can be found at Whole Foods and most grocery stores in the spice aisle. You can also order it on Amazon (my favorite resource for spices is Penzey’s Spices, they ship nationally).