Line two 8-inch pans with parchment paper, leaving extra paper hanging over the sides for easy removal.

In a medium bowl, whisk the condensed milk, peanut butter, vanilla, and half-and-half until smooth.

In another bowl, whip the heavy cream with a hand mixer or stand mixer until stiff peaks form.

Gently fold the peanut butter mixture into the whipped cream until fully combined and smooth.

Divide the ice cream mixture between the prepared pans and spread it into an even layer about 1/2 to 3/4 inch thick. Cover and freeze for 3 to 5 hours, or until firm.

Heat the oven to 375°F (190°C).

Beat the butter briefly, then mix in both sugars until light and fluffy.

Add the egg, then mix in the baking soda, salt, and vanilla.

Stir in the flour until a dough forms.

Scoop small portions of dough onto a nonstick baking sheet using about 2 teaspoons per cookie. Flatten each slightly with a fork or potato masher.

Bake for 8 to 10 minutes, until the edges are lightly golden.

Let the cookies rest on the tray for 5 minutes, then move them to a rack to cool completely.

Freeze the cooled cookies for 1 hour so they become firm enough for assembling.

Work in small batches to keep everything frozen. Take out one pan of ice cream and six cookies at a time.

Cut the ice cream into rounds with a 3- to 4-inch cutter, then sandwich each round between two cookies.

Press gently, then return the sandwiches to the freezer.

Repeat with the remaining ice cream and cookies. If the ice cream softens too much during assembly, freeze it again briefly before continuing.