Preheat oven to 350ºF. Crush the Oreos in a food processor until finely ground. Add the melted butter and mix again until all the crumbs are moistened. Press firmly into the bottom of a 9-inch springform pan. Bake for 8-10 minutes until set. Cool on rack.
In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese, sugar, and cornstarch together until smooth, scraping down sides of the bowl several times with a rubber spatula. With mixer on low speed beat in eggs, one at a time, then yogurt (or sour cream) and vanilla just until blended.
Divide the batter evenly between two medium bowls. Melt the chocolates separately. Stir semisweet chocolate into one bowl of batter and milk chocolate into the second bowl. Pour semisweet chocolate batter into prepared crust and spread an even layer. Carefully pour milk chocolate batter over semisweet chocolate layer and use an offset spatula to spread out evenly.
Bake for 50-55 minutes, or until set and center jiggles slightly when pan is shaken. Allow to cool completely in the pan on a wire rack.
In a stand mixer fitted with the whisk attachment (or with a beater), whip the heavy cream with the sugar until stiff peaks form. Transfer whipped cream to another bowl and return bowl to mixer. Beat the cream cheese until fluffy. Add the melted white chocolate and continue beating until smooth. Add the whipped cream and beat until combined. Spread mousse evenly on top of completely cooled cheesecake. Cover and refrigerate for at least 4 hours. Once set, garnish with optional chocolates.
When ready to serve, run a knife or metal spatula around the border of the cheesecake, then carefully remove the sides of the springform pan. Slice and serve!