In a small bowl, sprinkle the gelatin over 6 Tbsp. of cold water. Stir and set aside for 10 minutes to allow the gelatin to dissolve.
Meanwhile, in a large bowl whisk together 3 cups of the heavy cream, 4 cups of yogurt, 4 tsp. vanilla extract and the seeds from 2 vanilla beans.
Heat the remaining 3 cups of heavy cream and the sugar in a saucepan and bring to a simmer over medium heat.
Off the heat, stir the softened gelatin into the hot cream; stir until dissolved.
Next, pour the cream/gelatin mixture into the yogurt mixture; and 6 Tbsp. of rum (3 each of dark and spiced) and whisk until well combined.
Strain the mixture through a fine mesh sieve.
Pour into 12 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 4 tsp. rum (2 each dark and spiced) in a small saucepan over low heat; just until warm. Spoon a thin layer of warm caramel sauce on each cold panna cottage, sprinkle with salt and serve.