For the tea syrup: In a small saucepan, bring 1 cup of water to a boil. Add the tea bag to the hot water, and turn off the heat. Allow the tea to steep for about 10 minutes.
Remove the tea bag, and add the granulated sugar to the tea. Turn the heat back on to medium-high, and cook until the sugar has dissolved, a few minutes. Allow to cool completely. Once cooled, add the rum and set aside.
For the filling: In the bowl of a standing mixer fitted with the whisk attachment, beat the heavy cream until stiff peaks form. Transfer the whipped cream to another bowl, and set it in the fridge for a few minutes.
In the same mixer bowl, cream together the confectioners' sugar, mascarpone, pumpkin pie spice and vanilla for about 2 minutes on medium speed. Using a spatula, fold in the whipped cream, and set aside.
To assemble: In a 9-by-13-inch baking dish, place the gingersnaps in a single layer to cover the bottom. Carefully spoon about 1/2 cup of the tea syrup over them. Smear half of the filling on top, and repeat the process with the remaining gingersnaps (reserve 10), tea syrup and filling.
Pulse the reserved gingersnaps finely in a food processor. Sprinkle them over the last layer. Cover with plastic wrap, and chill for at least 6 hours before serving.
I usually make this as one big family-style dessert¿but occasionally I make it as individual desserts, served in mix-and-match dessert bowls or teacups. Just divide the tiramisu among 8 to 10 cups instead of using a baking dish.