Make the ice cream:

In a medium nonreactive saucepan over medium heat, combine the milk, heavy cream, sugar, and chocolate. Whisk until the sugar is dissolved and the chocolate melts completely. Heat the mixture to the scalding point (where bubbles form around the edge of the pan). Remove from the heat. Beat the egg yolks in a large mixing bowl. Whisk in 1/2 cup of the hot cream mixture and then whisk the egg mixture into the remaining hot cream mixture in the saucepan. Cook, stirring constantly, until the mixture is thick enough to coat the back of a spoon, 2 to 3 minutes.

Remove from the heat and strain through a fine-mesh sieve into a clean mixing bowl. Cover the top of the mixture with plastic wrap and transfer to the refrigerator until thoroughly chilled. Process in an ice cream machine according to manufacturer's directions. Transfer to an airtight, lidded container and freeze for 2 to 3 hours or up to overnight.

Make the cookies:

Heat the oven to 350 degrees F and position 2 oven racks in the center of the oven.

Combine the peanut butter, sugar, and egg in a mixing bowl and stir until smooth. Divide the dough into 12 portions, each about 2 heaping tablespoons each. Roll each portion in your hands to form a smooth ball. Place the cookie dough balls about 2 inches apart on ungreased baking sheets. Press down on each ball of dough with a fork to form a crosshatch pattern. Bake the cookies, switching the baking sheet positions mid way through cooking, until they are lightly golden, 12 to 14 minutes. Let the cookies cool on the baking sheets for 10 minutes before transferring to a wire rack to finish cooling.

When ready to assemble the ice cream sandwiches, place a heaping 1/2 cup of ice cream onto the center of 6 cookies. Top with 6 remaining cookies and press lightly to form a sandwich. Roll the edges of the cookies in the toasted nuts, if using, and serve immediately.