Preheat the oven to 350°F. Butter and flour a 12-cup bundt pan or liberally spray with baking spray. Sprinkle pecans evenly on bottom. Set aside.
Butter Rum Cake: Using an electric mixer cream together the butter, sugar and vanilla extract on medium-high speed in a large bowl until creamy, around 2 minutes. Add the eggs one at a time, beating well after each addition.
In a separate medium bowl use a whisk to sift together the flour, vanilla pudding mix, baking powder, baking soda and salt.
In a third bowl, whisk together the buttermilk and dark rum.
With the mixer on low speed gradually add dry ingredients alternately with the buttermilk mixture beginning and ending with flour. After all is added, beat just until fully combined stopping to scrape the sides of the bowl occasionally.
Pour cake batter into bundt pan. Bake for 50-60 minutes or until a toothpick or cake tester inserted into the center of the cake comes back clean.
About 10 minutes before the bundt cake is finished baking, make the buttered rum glaze.
Make the Butter Rum Glaze: In a medium saucepan, melt the butter over medium heat. Add the sugar, water and pumpkin pie spice. Bring to a boil, then lower to medium to simmer for 3 minutes, whisking occasionally. Remove from heat and add rum. Be careful the mixture may foam up. Continue to stir until foam dies down, placing back onto the stovetop with the heat turned off. Cook for 1 minute then set aside.
When cake is finished baking, remove from the oven. Use a long thin skewer to poke holes over the bottom of the cake. Slowly pour 1/2 buttered rum glaze over bottom of cake allowing it to soak in between pours. Let cake sit for 1 hour to cool in the pan on a wire rack.
After 1 hour, carefully invert cake onto a serving plate or stand. Use the same skewer to poke holes in the top of the cake. Try to poke between the pecans. Slowly pour remaining glaze into holes on top of cake. Be patient this will take several minutes. The slower you go, the more glaze goes into the cake and doesn't pool on platter. (It may be necessary to warm the glaze slightly before beginning the process of pouring on the top.)
Let cool completely to room temperature. Once cooled, cover and chill for at least 1 day. The rum flavor mellows the longer this cake proofs.