Make the filling: Bring the milk to a simmer in a medium saucepan over medium heat. Whisk the granulated sugar, cornstarch, peppermint extract, egg and egg yolk in a medium heatproof bowl until smooth. Gradually whisk half of the hot milk into the sugar mixture, then return to the pan with the remaining milk. Reduce the heat to medium low and cook, whisking constantly, until the mixture boils and thickens, 2 to 3 minutes. Transfer to a bowl (strain through a fine-mesh sieve if it looks lumpy), then stir in the white chocolate and green food coloring until the chocolate melts. Press plastic wrap directly onto the surface and refrigerate until cold, at least 1 hour. Stir in the chocolate chips.

Make the topping: Beat the heavy cream, confectioners' sugar and peppermint extract in a bowl with a mixer on medium speed until medium peaks form, 1 to 2 minutes.

Assemble the cake: Put 1 cake layer on a platter; spread the filling on top, stopping 1/2 inch from the edge. Top with the second cake layer and press down gently. Top with the whipped cream and chocolate chips.

Photograph by Levi Brown