In a glass bowl, melt the unsalted butter (½ cup, i.e., 1 stick), dark chocolate chips (½ cup), and semi-sweet chocolate chips (½ cup) in the microwave on 50% power for 1 minute. Stir mixture and continue to melt in 20-second increments in the microwave until butter and chocolate are melted & smooth. Allow to cool for 2 minutes.
In a large mixing bowl, mix together the chocolate/butter mixture, unsweetened cocoa powder (3 tablespoons), granulated sugar (¼ cup), and light brown sugar (¼ cup) with an electric mixer for 30 seconds. Add egg (1 large) and stir until combined.
On low speed, add in baking soda (1 teaspoon) and stir for 10 seconds. Slowly add in all-purpose flour (2 cups) and mix for 30 seconds, or until dough is combined.
Scoop by the tablespoon and roll into balls. Place each ball on a parchment-covered baking sheet. Flatten each ball to ¼ inch thick and using a 2 inch diameter cup, cut a circle into each pressed cookie. Remove excess dough after cutting and throw back into the ball to remix.
Bake at 350℉ for 8 minutes. Allow to cool on the hot pan for 10 minutes before removing cookies from the tray.
To make filling, beat the cream cheese (4 ounces), softened unsalted butter (¼ cup, i.e., ½ stick), vanilla extract (½ teaspoon), and melted white chocolate chips (¼ cup) on high speed with an electric mixer for 30 seconds. Add in powdered sugar (½ cup) and continue to beat for another 30 seconds. Pipe 1 tablespoon of filling between each "pair" of cooled cookies.
If not serving immediately, store in airtight container in the fridge for up to 3 days.