For the cake:
Preheat oven to 350 degrees F. Spray 3 (8-inch) round cake pans with nonstick butter-flavored baking spray; set aside.
In a large bowl, beat cake mix, rum, water, eggs, and oil on low speed with an electric mixer for 30 seconds. Scrape down sides of bowl and beat for 2 more minutes on medium speed.
Divide batter evenly between prepared cake pans. Bake in preheated oven for 20 to 25 minutes, or until tester comes away clean. Cool completely before icing.
For the icing:
In a medium saucepan over medium-low heat, melt butter with brown sugar and rum extract. Bring to a boil, stirring constantly. Add milk and bring to a hard boil. Stirring constantly, until sugar is completely dissolved; about 2 minutes.
Let cool to room temperature and transfer to a medium bowl. Use an electric mixer on medium speed to beat in powdered sugar until smooth. If too thick, add a little milk. If too thin, add a little powdered sugar.
Using an offset spatula, evenly spread icing on top of each cake round. Stack and ice the top of the cake. You may have frosting leftover.