Combine the sugar and 1/2 cup water into a pot and stir to combine, but do not stir from this point forward. Set over medium to medium-high heat depending on your stovetop. Without stirring, cook the mixture until dark amber in color, swirling the pan carefully while cooking, about 5 to 10 minutes. I like to watch the pot the entire process as it can burn so quickly.
Slowly add the cream and remove from heat, being careful because the caramel can splatter up. Stir with a wooden spoon. Add the butter. Add the vanilla extract and rum and keep stirring. The caramel needs to settle down. Set aside.
When the cake is ready and done, remove from the oven and cool on a wire rack for 6 to 8 minutes.
You can gently loosen the sides of the cake with a pairing knife. Slowly remove the cake to the platter by putting the platter top down on the top of the cake pan and quickly turn over, so the cake comes out; pour caramel topping slowly over the warm cake.
Finally, poke small holes in the cake with a bamboo skewer. Pour the caramel sauce over the cake and into the holes.
When ready to serve, slice the cake, the thickness you desire. Place on a plate. It can be served with vanilla ice cream or flavored whipped cream.