In a small saucepan over medium heat, stir together brown sugar and butter. Allow the butter to melt and the sugar to dissolve. When mixture is bubbly, whisk in 1/2 cup of the heavy cream until smooth. Remove from heat and stir in vanilla extract and dark rum.
In a large saucepan over medium high heat, combine the remaining heavy cream with the whole milk and bring to a low simmer, but do not allow the mixture to boil.
In a mixing bowl, beat the egg yolks. Drizzle approximately 1/2 cup of the heated milk and cream mixture into the egg yolks while whisking constantly, and then pour back into the simmering saucepan.
Reduce heat to low and stir constantly until the mixture is slightly thickened and coats the back of the spoon. This will take 4-6 minutes. Remove from heat.
Pour mixture through a fine mesh strainer into a clean bowl and then stir in the butterscotch mixture. Cover with plastic wrap, allowing the wrap to touch the surface of the mixture, and chill at least 3 hours or overnight.
While the mixture is chilling, bring cream soda to a boil over medium high-heat, and then reduce to a heavy simmer. Allow to simmer until thickened and reduced to 1/3 of the original amount. Remove from heat and allow to cool to room temperature or transfer to a container and chill in the refrigerator.
Once ice cream base has thoroughly chilled, churn in ice cream maker according to the manufacturer's instructions. Layer 1/3 of the ice cream mixture in the bottom of a freezer safe container and drizzle on some of the cream soda syrup. Repeat layers until all of the ice cream is used. If desired, reserve some of the syrup to drizzle on when ice cream is served.