Line an 8x8 or 9x9-inch square baking dish with parchment paper, making sure the edges hang over the sides for easy removal later.
In a microwave-safe bowl, combine the semi-sweet chocolate chips, half of the sweetened condensed milk, and 1 tablespoon of butter.
Heat in the microwave for 30 seconds, then stir until the mixture begins to come together. If needed, microwave for an additional 15 seconds and stir again until completely smooth. Stir in 1 teaspoon of vanilla extract.
Pour the chocolate mixture into the prepared pan and spread it evenly to form the first layer. Place the pan in the refrigerator for about 30 minutes to set.
For the second layer, combine the white chocolate chips, remaining sweetened condensed milk, and 1 tablespoon of butter in a clean microwave-safe bowl. Heat for 30 seconds, stir, and microwave another 15 seconds if necessary. Stir until smooth, then mix in the remaining 1 teaspoon of vanilla extract.
Carefully spread the white chocolate mixture over the chilled chocolate layer, smoothing the top with a spatula.
Immediately add the Christmas sprinkles over the top before the fudge begins to set.
Refrigerate the layered fudge for at least 4 hours, or overnight, until firm.
Once set, lift the fudge out of the pan using the parchment overhang, cut into squares, and serve.