In a small bowl, sprinkle the gelatin on 3 tablespoons of cold water. Stir and set aside for 10 minutes to allow the gelatin to dissolve.
Meanwhile, in a large bowl, whisk together 1 1/2 cups of the cream, the yogurt, vanilla extract, and vanilla seeds. Heat the remaining 1 1/2 cups of cream and the sugar in a small saucepan and bring to a simmer over medium heat. Off the heat, stir the softened gelatin into the hot cream until dissolved. Pour the hot cream-gelatin mixture into the cream-yogurt mixture and stir in 3 tablespoons of rum. Pour into 8 serving glasses and refrigerate uncovered until cold. When the panna cottas are thoroughly chilled, cover with plastic wrap and allow to chill overnight.
Before serving, heat the caramel sauce with 2 teaspoons of rum in a small bowl in the microwave or a small saucepan set over low heat, just until warm. Spoon a thin layer of warm caramel on each cold panna cotta, sprinkle with fleur de sel, and serve.