Heat the oven to 350 degrees. Butter a 9x2-inch round cake pan. Line pan with parchment, and butter the parchment. Coat pan with flour, and tap out any excess. In a large bowl, sift together flour, cinnamon, nutmeg, allspice, salt, baking powder, and baking soda; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat sugar and butter together until light and fluffy, about 2 minutes. Add eggs, and beat until combined. Add pumpkin purée and milk; beat until combined. Add reserved flour mixture; beat on low speed until just combined.
Pour batter into prepared pan, and bake until a cake tester inserted into the middle comes out clean, about 55 minutes. Transfer the cake to a wire rack to cool. Let cake rest 20 minutes. Unmold cake. Using an offset spatula, spread icing over top of cake, and decorate with caramelized walnut halves.
To make Brown Butter Icing, in a small saucepan, melt butter over medium-high heat until nut-brown in color, about 10 minutes (see note). Remove pan from heat, and pour butter into a bowl, leaving any burned sediment behind. Add sugar, vanilla, and 1 Tbs. milk; stir until smooth. If the icing is too thick, add the remaining Tbs. milk, a little at a time, until consistency is spreadable. Let cool 5 minutes. Use immediately.
Note: Browning the butter in the frosting gives this icing a nutty flavor. The consistency should be thin enough to spread but not runny.
Yield: 8-10 servings