For the cookies:
Preheat an oven to 350 degrees F.
In a metal bowl, combine both chocolates with the butter. Put the bowl over a pot of boiling water for a makeshift double boiler. Lower the heat so the water is hot but not boiling. Stir the chocolate with a plastic spatula, from time to time, until the chocolate and butter melt together. Set aside.
In a large bowl, whisk together the egg, sugar and vanilla. Beat until the mixture comes together and becomes a pale yellow color, about 1 minute.
Combine the flour, baking powder and salt in a separate bowl. Sift and add it to the egg mixture. Stir until blended. Using a rubber spatula, add half of the melted chocolate to the batter and gently stir to combine. Add the remaining chocolate and stir again until combined.
Drop tablespoonfuls of the cookie batter onto a greased baking sheet or a greased parchment-lined baking sheet. Leave room in between the cookies so they can spread out as they bake. You should be able to drop 18 cookies total, on the baking sheet, depending on their size.
Put the baking sheet in the center of the oven and bake for 4 minutes. Rotate the pan halfway and bake for an additional 4 minutes. Remove the pan from the oven and allow the cookies to cool, 10 to 15 minutes, before transferring to a plate.
For the filling:
Preheat the oven to 250 degrees F. Put the chopped white chocolate into a baking dish, preferably not a metal dish, and melt it in oven. Remove it after about 10 minutes and stir. Put back into the oven and continue to cook until caramelized, about 10 minutes.
Meanwhile, whisk together the mascarpone and rum in a small bowl.
Add the caramelized chocolate to a medium bowl. Blend in the mascarpone mixture until combined. Refrigerate until the cookies have cooled.
Bring the filling to room temperature before spreading it between the cookies to make sandwiches.