For the cake: Preheat the oven to 350 degrees F. Lightly grease a 9-by-13-inch baking pan with canola spray.
Prepare the boxed cake batter according to the package directions, adding the peppermint extract. Spread evenly in the prepared pan and set aside.
Combine the ricotta, mascarpone, sugar, eggs and vanilla in the bowl of a stand mixer fitted with the paddle attachment. Beat until smooth. Gently scoop the filling onto the cake batter and spread so the top is completely white.
Bake until a toothpick inserted into the center of the cake comes out clean, 50 to 65 minutes. Remove from the oven and let cool 20 minutes.
For the ganache: Combine the chocolate chips, cream, corn syrup and peppermint extract in the top of a double boiler. Gently stir the mixture until the chocolate melts and the ganache becomes very smooth.
Spread the ganache evenly over the cooled cake and let stand 10 minutes to set. Cut the cake into 18 squares.