Substitute extra cream for rum in the sauce, if you like. For a thicker glaze, add the whole 1 1/3 cups powdered sugar, whisking constantly, as the glaze begins to cool in the pan. For a sheer, saucy glaze, use less powdered sugar.
Prepare Cake: Preheat oven to 325°. Beat 2 cups butter at medium speed with a heavy-duty electric stand mixer until creamy. Gradually add 3 cups brown sugar, beating until light and fluffy. Add eggs, 1 at a time; beat until blended after each addition.
Stir together flour and next 5 ingredients. Add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture. Beat at low speed just until blended. Stir in pecans and vanilla. Spoon batter into a greased (with shortening) and floured 15-cup Bundt pan.
Bake at 325° for 1 hour and 15 minutes or until a wooden pick inserted in center comes out clean. Cool in pan on a wire rack 15 minutes; remove from pan to wire rack, and cool completely.
Prepare Glaze: Bring 1/4 cup brown sugar and next 3 ingredients to a boil in a small saucepan over medium heat, whisking constantly; boil, whisking constantly, 1 minute. Remove from heat, and whisk in rum. Add 1/3 cup powdered sugar; whisk until smooth. Gradually add more powdered sugar, whisking constantly, until desired thickness is reached. Drizzle glaze over cake.