Wrap an 8-inch springform pan in tinfoil. Preheat oven to 350 degrees.
Place the cookie parts of the Oreo cookies and the sugar in the bowl of a food processor. Pulse until the mixture resembles fine crumbs. Add the melted butter, and pulse a few times until all the crumbs are moistened.
Press the crumbs into the bottom of the springform pan and about half an inch up the sides. Bake for 10 minutes, then remove to a wire rack to cool.
Make the filling: Beat the cream cheese, sugar and cornstarch in the bowl of a stand mixer fitted with the paddle attachment on medium speed until smooth, scraping down the sides and bottom of the bowl several times with a silicone spatula. With the mixer on low speed beat in the eggs, 1 at a time, scraping the sides and bottom of the bowl before adding the next egg. Add the sour cream and vanilla and beat until medium-low just until blended.
Divide the batter evenly (about 2 cups each) among 3 medium bowls. Stir the semisweet chocolate into 1 bowl of batter, white chocolate into another and milk chocolate into the third. Spread the semisweet chocolate batter onto the prepared crust in an even layer. Carefully spoon the white chocolate batter over bittersweet chocolate layer to cover; gently smooth with an offset spatula (do not let batters stir together). Spoon the milk chocolate batter over the white chocolate layer to cover; smooth with an offset spatula.
Bake for 1 1/4 hours or until set and center still jiggles slightly when gently shaken (cake should have pulled away from sides of pan; if not, carefully run a thin knife around edge of pan to release cheesecake). Cool in pan on a wire rack. Cover; refrigerate at least 4 hours or up to 2 days.
Chocolate Glaze: Remove the side of the springform pan. Place the chopped chocolate in a small bowl. Warm the heavy cream in a small saucepan over medium heat to scalding (small bubbles form at the edge of the pan). Pour the hot cream over the chocolate and let stand for 1 minute, then stir until smooth. Pour the glaze over the cheesecake, allowing it to drip down the sides.
Refrigerate just until glaze is set or, if chilled longer, let stand at room temperature 30 minutes before serving.