In a nonstick skillet, melt the butter. Add the bananas in a single layer and sprinkle with the brown sugar. Cook over moderate heat, turning once, until caramelized, 8 minutes. Off the heat, add the rum and swirl the pan to dissolve the sugar. Scrape three-quarters of the bananas into a food processor and add 3 tablespoons of the milk. Puree until smooth. Transfer the puree to a small bowl and freeze until chilled, 15 minutes. Chop the remaining bananas and freeze until chilled.
In another bowl, whisk the cocoa with the granulated sugar, salt, vanilla and the remaining 1/2 cup of milk. Whisk in the yogurt until smooth, then the banana puree.
Transfer the mixture to an ice cream maker and freeze according to the manufacturer's instructions until nearly frozen. Mix in the chopped bananas and chocolate. Scrape the frozen yogurt into an airtight container, cover and freeze until firm, at least 2 hours.