In a food processor, pulse Oreos until finely ground (or you can use a large Ziploc bag and rolling pin, then pour into a mixing bowl and toss with butter), with mixer running slowly pour in melted butter and pulse until evenly coated. Pour about 2 1/2 Tbsp of the Oreo mixture into 6 serving cups and level, set aside.
Melt white chocolate according to directions listed on package (I did the microwave directions), then allow to cool until just barely lukewarm (keep if fluid though, you don't want it to set at all or you'll get pieces of white chocolate throughout the cheesecake).
In a medium mixing bowl, using an electric hand mixer, whip heavy cream until soft peaks form. Add 3 Tbsp powdered sugar and whip until stiff peaks form, set aside in refrigerator.
Meanwhile, in a mixing bowl using an electric hand mixer, whip cream cheese until smooth and fluffy. Mix in remaining 5 Tbsp powdered sugar and 1/2 tsp peppermint extract. Stir in cooled white chocolate (at this point if you want to tint half with red food coloring then separate half into another bowl and tint then add half of the whipped cream to each) then fold in whipped cream. Spoon mixture into a piping bag (if doing pink and white, follow directions here). Pipe over oreo layer in cups. Chill in refrigerator 2 hours.
Garnish with either crushed candy canes, candy canes sticks (you'll only need bottom half of candy canes), or peppermint white chocolate dipped mini Oreos just before serving.
If doing the peppermint white chocolate dipped Oreos, simply melt some white chocolate or vanila almond bark according to directions on package and dip mini Oreos in melted white chocolate. Transfer to wax paper and sprinkle with crushed candy canes before chocolate sets. Allow to set at room temperature.