Preheat the oven to 350°F with a rack in the center. Line an 8" square pan with a parchment sling, then lightly grease the parchment.
To make the crumbs: In a medium bowl, stir together the butter, sugars, and salt. Sift in the flour and cocoa powder, then stir until a smooth, soft dough forms. Cover and set aside while you make the cake batter.
To make the cake: In a large bowl, vigorously whisk the sugar and egg until pale and thick, about 1 minute. Add the oil and whisk until thick, shiny, and emulsified.
Whisk in the pumpkin, vanilla, spice, baking powder, baking soda, and salt until combined. Add the flour and whisk until no dry streaks remain.
Transfer the batter to the prepared pan and spread it into an even layer.
Use your hands to break the dough for the crumbs into pieces of various sizes (some as big as almonds, others as small as chickpeas and lentils), scattering them evenly over the cake batter.
Bake for 28 to 32 minutes, until a toothpick or paring knife inserted into the center of the cake comes out clean; the internal temperature should be at least 200°F when measured with a digital thermometer.
Remove the cake from the oven and let it cool in the pan for about 20 minutes. While the cake is cooling, prepare the glaze.
To make the glaze: In a small bowl, stir together the glaze ingredients until smooth, starting with smaller amount of milk. Add additional milk as needed until the glaze is the consistency of molasses.
Drizzle the glaze all over the surface of the cake, allowing it to flood the surface. Let the glaze set for 30 to 45 minutes before removing the cake from the pan, using the parchment sling as handles. Slice and serve at room temperature.
Storage information: Store leftover pumpkin crumb cake, well wrapped, at room temperature for several days; freeze for longer storage.