In a large skillet over medium heat, melt butter completely.
Add brown sugar, cinnamon, and a pinch of salt. Stir constantly until the sugar dissolves into the butter and the mixture is smooth and bubbly, about 2 minutes.
Add banana slices to the pan in a single layer. Cook for 1-2 minutes per side until they begin to soften and caramelize, spooning sauce over them gently.
Remove the pan from heat. Carefully pour in the rum and vanilla extract.
Return pan to heat. If desired, carefully tilt the pan toward the flame to ignite the rum for a flambé (optional). If not flambéing, simply let the sauce simmer for 2-3 minutes until the alcohol cooks off.
Cook for another minute until the sauce thickens slightly and coats the bananas beautifully.
Scoop vanilla ice cream into bowls and spoon the warm bananas and sauce over the top. Serve immediately.