Line a 9×13-inch baking pan with foil, with ends extending over sides. Finely crush 24 Oreo cookies. Melt 1/4 cup butter; mix with crumbs. Press onto bottom of prepared pan.
In a large bowl, beat the cream cheese and sugar with mixer until blended. Add sour cream and vanilla; mix well. Add eggs, one at a time, beating after each just until blended. Chop remaining 12 cookies. Gently stir into batter; pour over crust.
Bake the cheesecake for about 35-40 minutes or until the sides are set and the center is almost set. Cool completely on a wire rack in the pan.
When the cheesecake is completely cooled, cover with plastic wrap and refrigerate for at least 2 hours.
Once chilled, remove the cheesecake using the foil overhang and cut the cheesecake into bite-sized pieces. Place the cheesecake bites on a wax or parchment paper-lined baking tray. Melt the semisweet (or milk) chocolate and white chocolate in separate bowls (can microwave on 50% power, stirring frequently).
Pour the melted chocolate into a ziploc bag, one for the semisweet chocolate, one for the white chocolate. Snip a small corner off the corner of the bag and drizzle the chocolate over the cheesecake bars, alternating with each.