In a medium, heavy saucepan, heat the cream, milk, mint, and ½ cup of the sugar over medium heat, stirring occasionally, until almost simmering. Turn off the heat, cover the pan, and steep the mint in the cream for 15 minutes. In a medium bowl, whisk together the egg yolks, the remaining ½ cup sugar, and the salt. Slowly pour the hot liquid into the eggs, whisking as you pour. Return the mixture to the pan and cook over medium-low heat, stirring constantly with a heat-resistant plastic or wooden spatula, until it reaches 175 degrees Fahrenheit and lightly coats the spatula. Strain the mixture into a clean bowl, discarding the mint. Cool over an ice bath until room temperature. Refrigerate for at least 4 hours or up to overnight.
In a food processor, using on-off pulses, grind the chocolate into small pieces. Put the chocolate in a bowl in the freezer. Churn the ice cream base in an ice cream machine according to the manufacturer's instructions. Place the ice cream in the bowl with the chocolate and fold the two together. Freeze until scoopable, about 4 hours, depending on your freezer.
Preheat the oven to 350 degrees Fahrenheit. Line 3 baking sheets with parchment paper. Melt the butter and chocolate together in a double boiler over hot water. Stir to combine. Let cool to room temperature. Stir together the flour and baking powder. Add the salt. With an electric mixer on high speed, whip the eggs, sugar, and vanilla extract together until thick. By hand, stir in cooled chocolate mixture. Stir in the flour mixture and let the batter rest for 5 minutes. Using an ice cream scoop 2 ¼ inches in diameter for each cookie, scoop the batter onto the prepared pans at least 2 inches apart. There should be at least 20 cookies. Bake for about 10 minutes, or until the tops crack. They will look set and no longer shiny on top. Let cool and then remove them form the pans with a spatula.
Assemble the sandwiches
Place 10 of the cookies, bottom side up on a baking sheet lines with parchment or waxed paper. Place a scoop on ice cream on top of each cookie. Top with a second cookie and gently press down on the top cookie to adhere the sandwiches together. Serve immediately, or freeze until ready to serve.