Using a microwave-safe bowl, MELT chocolate until smooth (30 second increments, stirring in between). Let cool.
In the bowl of your mixer fitted with the paddle attachment, COMBINE the cream cheese and sugar. Beat together until smooth.
ADD in the cooled, melted chocolate and mix until combined. One at a time, ADD peppermint and eggs, making sure to scrape down the bowl in between each egg.
DROP batter into prepared crust. FILL a larger pan (10x10 cake pan or roasting pan) with water halfway for a water bath. PLACE the cheesecake pan into the water bath.
BAKE 45 to 55 minutes until the edges appear to be set. The center will still have some jiggle to it. Close the oven door, turn the heat off, and let rest in the cooling oven for 1 hour.
After 1 hour has passed, REMOVE the cheesecake from the water bath and place on a cooling rack to cool completely. Once the cake is completely cooled, CHILL for at least 5 hours or overnight.