Preheat oven to 350 degrees. Grease a 12-cup bundt pan and set aside.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, allspice, and black pepper. Set aside.
Add brown sugar, butter, and olive oil to the bowl of a stand mixer fitted with the paddle attachment. Mix on medium speed until light and fluffy, about 3 minutes. Add the eggs, one at a time, beating for about 20 seconds after each to combine. Add pumpkin purée and sour cream; mix until well combined. Scrape down the sides of the bowl.
Remove bowl from mixer and use a rubber spatula to fold in the dry ingredients, 1/4 at a time. Be sure to scrape the sides and bottom of bowl to ensure batter is fully combined (but not over-mixed).
Pour batter into prepared pan, smoothing out the surface. Tap the pan on the counter a few times to release any large air bubbles. Bake on center rack for 50 - 55 minutes, or until a toothpick inserted in the center comes out clean.
Set the cake, in its pan, on a wire rack to cool for 20 minutes. After 20 minutes, invert the cake onto the rack to cool completely.
Once the cake has cooled, make the icing. Begin by melting butter in a small saucepan over medium-low heat. Use a rubber spatula to scrape the sides as it cooks, watching it closely. Once butter begins to turn light brown and has a nutty aroma, remove from heat and transfer to a heat-safe bowl. Allow to cool slightly.
Add confectioners' sugar, maple syrup, and salt to brown butter and whisk until smooth. Icing should be thick, but pourable. If it is too thin, add more confectioners' sugar. Transfer cake to serving platter and pour icing evenly over top. If desired, use a knife or spatula to smooth out the icing on the top of the cake. Serve in generous slices.