Sprinkle the gelatin over the rum in a small teacup. Set aside for 5 minutes while you continue with the recipe.
In a small saucepan, combine one third of the heavy cream and the sugar. Stir on medium-low heat to gently warm the cream and dissolve the sugar. You can tell when the sugar is dissolved by the sound and the feel of the spoon while stirring the bottom of the saucepan. When the sugar is dissolved, add the rum and gelatin mixture, and stir until the gelatin dissolves and disappears.
Remove the mixture from the heat, add the pumpkin purée and the vanilla. Stir (or whisk gently if necessary) to combine, and be sure to get rid of any lumps. Strain the mixture through a fine sieve into a medium bowl. Set aside to cool to room temperature.
When the pumpkin mixture is cool, whip the rest of the cream in the bowl of an electric mixer fitted with the paddle attachment. Fold the pumpkin mixture into the whipped cream, divide the mixture among 4 glasses, Mason jars, or teacups. Refrigerate them to set, at least 3 or 4 hours before serving.
Serve the mousse with chocolate spice cookies, or crumble them over top before serving. You could also top each with a dollop of whipped cream.