In a medium-sized bowl, whisk together the sweetened condensed milk, pure vanilla extract, peanut butter, and half-and-half, until the mixture is smooth. Set this aside. With a standing or hand mixer, whip the heavy cream until stiff peaks form, 60-90 seconds. Fold the sweetened condensed milk mixture into the cream that has been whipped. This is your ice cream base. You will want this to be as smooth as possible and lump-free.
Unwrap the mini peanut butter cups. Place them in a large plastic bag and seal it. Gently roll a rolling pin over the peanut butter cups to crush them into chunks, being careful as they are a softer candy. Alternatively, you can crush them by hand, roughly breaking them into four pieces per cup. For larger chunks of peanut butter cups, simply crush them less. For smaller chunks of peanut butter cups, crush them more. Place the crushed peanut butter cups into the ice cream base and stir to combine, ensuring that there is an even distribution of cups throughout the ice cream batter.
Pour the ice cream into an 8 x 8–inch baking pan (or a freezer-safe container of about the same size), and using your spatula, spread the ice cream evenly throughout the pan. Freeze the ice cream uncovered for 3-5 hours or overnight, until it is firm. Cover any leftovers with aluminum foil or plastic wrap, and store them in the freezer for up to 3 weeks.