For the mint chocolate chip ice cream:

Prepare an ice bath by filling a large bowl with ice cubes and 1 to 2 cups of water. Place a medium bowl fitted with a fine strainer inside the ice bath.
In a medium saucepan, combine the milk, cream, 1/2 cup sugar, and salt. Set over medium heat, stirring occasionally, until the mixture is warm and begins to steam, about 5 minutes.
In a medium bowl, whisk together the egg yolks and the remaining 1/4 cup sugar. Carefully whisk half of the warm milk mixture into the egg yolks, one ladleful at a time, until the egg mixture is warmed. Whisk the egg-milk mixture back into the saucepan. Cook the mixture over medium heat, stirring constantly with a wooden spoon, until the mixture is thick enough to coat the back of the spoon and registers around 175°F on an instant-read thermometer, about 5 to 7 minutes. Be careful not to boil the mixture.
Immediately strain the mixture through the fine strainer into the prepared ice bath. Stir in the extracts and food coloring, if using. Cool the custard in the ice bath until it reaches room temperature, stirring often. Press plastic wrap against the surface of the custard and refrigerate until chilled, about 4 hours or up to 1 day.
Pour the chilled custard into an ice cream maker and freeze according to the manufacturer’s directions. Add the chocolate chips during the last 5 minutes of churning. Transfer the ice cream to an airtight container, press plastic wrap against the ice cream surface, and freeze until the ice cream is firm and the flavor is ripened, at least 2 hours.
For the chocolate sugar cookies:
Preheat the oven to 350°F. Line large baking sheets with parchment paper or silicone baking mats.
In a medium bowl, sift together the flour, cocoa powder, baking soda, baking powder, and salt.
In a large bowl, use an electric mixer to beat the butter and 1 cup of the sugar on medium-high speed until well combined and smooth. Beat in the eggs and vanilla. On low speed, add the flour mixture and beat until combined.
Place the remaining 1/2 cup sugar in a shallow dish. Roll the dough into 2 tablespoon-sized balls, then roll in the sugar. Place on prepared baking sheets and flatten with the bottom of a measuring cup to a 2-inch diameter.
Bake for about 10 to 12 minutes, or until the cookies are set. Let cool on baking sheets for 5 minutes before removing to wire racks to cool completely. Freeze the cookies until firm, at least 1 hour. The cookies can be stored in airtight containers in the freezer for up to 1 month.
For assembly:
Top one cookie with a scoop of ice cream. Place another cookie on top of the ice cream and gently press down to form a sandwich. Immediately place the sandwich in the freezer. Repeat for the remaining cookies. Freeze for at least 1 hour before serving.