Combine the bananas and sugars in a medium pot. Mash with a fork until the fruit is chunky and the sugar syrupy. Set over medium-low heat and bring to a simmer. Add the lime juice, water, and salt. Continue to cook at a steady simmer—stirring occasionally—for about 10 minutes, until slightly thickened. Remove from the heat and stir in the rum and vanilla.
Spoon the hot jam into jars. I just seal with a lid and keep in the fridge—but sterilize if you'd like.
Emma is a writer and recipe developer at Food52. Before this, she worked a lot of odd jobs, all at the same time. Think: stir-frying noodles "on the fly," baking dozens of pastries at 3 a.m., reviewing restaurants, and writing articles about everything from how to use leftover mashed potatoes to the history of pies in North Carolina. Now she lives in Maplewood, New Jersey with her husband and their cat, Butter. Stay tuned every Tuesday for Emma's cooking column, Big Little Recipes, all about big flavor and little ingredient lists. And see what she's up to on Instagram and Twitter at @emmalaperruque.