Line an 8-inch square pan with foil and grease the foil with butter. (I like to do this by putting my hand in a plastic sandwich bag like a glove and then dipping the bag in some butter. Then when you are done spreading it around on the foil, you can just throw the plastic bag away!) Sprinkle the pan with half of the crushed candy canes.
Combine the chocolate chips and 2 Tbsp butter in a microwave safe bowl and microwave for 1 minute. Stir and then microwave at 15 second intervals, stirring until smooth. (Be careful not to microwave this for too long - mine only took about 1 minute and at that point it didn't look completely melted but when I stirred it, it became smooth.)
Stir in 2/3 cup of the condensed milk until well mixed and then pour over the crushed candy canes in the pan. Refrigerate for about 10 minutes.
After 10 minutes, combine the white chocolate and 2 Tbsp butter in a microwave safe bowl. Microwave for about 45 seconds and then stir until smooth.
Stir in the cream of tartar, the remaining sweetened condensed milk and the remaining crushed candy canes. Spread this over the chocolate layer.
Refrigerate for about 2 hours or until firm. Using the foil, remove the fudge from pan and invert onto a cutting board to cut the fudge into small pieces. Enjoy!