Make the cakes: Preheat oven to 325 degrees. Coat three 8-by-2-inch round cake pans with cooking spray, line with parchment, and coat parchment. Whisk flour, baking powder, baking soda, and salt in a medium bowl.
Beat butter and brown sugar with a mixer on medium-high speed until pale and fluffy, about 5 minutes. Beat in eggs, 1 at a time, beating well after each addition, then add vanilla and rum. Reduce speed to low. Add flour mixture in 2 additions, alternating with buttermilk. Raise speed to medium-high, and beat for 2 minutes. Divide batter among pans.
Bake cakes until golden brown and testers inserted in centers come out clean, about 40 minutes. Transfer pans to wire racks, and let cool slightly. Invert cakes onto racks. Let cool.
Make the frosting: Melt 1 stick butter in a medium saucepan over medium heat until dark golden brown, about 10 minutes. Add brown sugar, cream, and salt, stirring until sugar dissolves. Bring to a boil, whisking constantly, and cook for 3 minutes. Transfer to a mixer bowl, and let cool.
With machine running, add remaining butter, a few pieces at a time, and beat on low until incorporated. Raise speed to medium, and beat for 2 minutes. In another bowl, beat cream cheese and confectioners' sugar on medium-high until fluffy and smooth, about 3 minutes. Add brown-butter mixture to cream cheese, and beat until smooth. Cover, and refrigerate until chilled, at least 2 hours (or overnight, beating on low speed before using).
Make the butterscotch sauce: Mix sugar, butter, corn syrup, and salt in a small saucepan over medium heat, and cook, stirring, until sugar dissolves. Bring to a boil, and cook for 2 minutes. Remove from heat, and whisk in cream. Return to heat, and cook for 2 minutes. Let cool slightly.
To assemble the cake: Trim tops of 2 cake layers and top and bottom of third (this will be the middle) to create flat exposed surfaces. Brush 1/4 cup plus 2 tablespoons butterscotch sauce on cut sides. Let cool. Spread 1 cup frosting on 1 layer, then place middle layer on top, sauce side down. Spread 1 cup frosting on middle layer, then place third layer on top, sauce side down. Spread 1 cup frosting on top and sides. Refrigerate until firm, about 1 hour.
Using an offset spatula, spread remaining frosting on top and sides of cake. Press chopped pecans on sides, and garnish top with halves. Transfer cake to a serving plate or cake stand. Refrigerate until firm, at least 4 hours (or up to 2 days, covered).