Combine the sweetened condensed milk, peppermint extract, and green food coloring in a large mixing bowl. Set aside.

Place heavy cream into the bowl of a stand mixer fitted with the whisk attachment (or into a large mixing bowl and use an electric mixer). Beat the heavy cream on medium speed until it holds stiff peaks.

Fold about half a cup of the whipped cream into the sweetened condensed milk mixture to lighten. Fold in the remaining heavy cream until just combined, then fold in the chocolate chips.

Cover loosely with plastic wrap and freeze for at least 4 hours, or overnight.

Preheat the oven to 350 degrees.

Place the cake mix, eggs, oil, and peppermint extract in a large mixing bowl and beat with an electric mixer set to medium speed until ingredients are well combined, 2 to 3 minutes. Fold in the chocolate chips. (Dough will be very stiff.)

Form into balls of about 2 tablespoons of dough. Place on ungreased baking sheets 2 inches apart and flatten slightly. Bake until lightly browned around the edges, 10 to 12 minutes. Cool completely on baking sheets before filling with ice cream. (You can make the cookies up to 1 week ahead and store in an airtight container at room temperature.)

To make the ice cream sandwiches, place a scoop of ice cream on half of the cookies, then top the ice cream with another cookie and press down to form a sandwich. Roll the edges of the ice cream in additional mini chocolate chips, if desired. Freeze sandwiches for 30 minutes, then serve.