In a saucepan over medium low heat, whisk together the heavy cream, half and half, sugar, and vanilla bean paste. Stir occasionally, heating the mixture until bubbles form around the edges. Remove from heat.
While cream is heating up, whisk 6 egg yolks together until frothy and slightly pale yellow (I like to use my hand mixer to do this, it makes it way faster). After removing the cream mixture from heat, remove about 1/4 to 1/3 cup of the hot cream mixture and vigorously whisk into the eggs. Place the pan of cream back on the stovetop over medium low heat, and whisk the egg mixture back into the remaining cream. Heat, stirring constantly, until thickened (it only takes a couple minutes). You should be able to dip the back of a spoon into the mixture and drag your finger through the cream on the back of the spoon and it should leave a trail. Remove from heat place a piece of saran wrap up against the custard mixture and chill in the refrigerator (for several hours or overnight).
Once the mixture is chilled freeze according to your ice cream maker’s instructions. When the custard is almost done churning, toss in the chopped peanut butter cups into the ice cream machine. Once those are mixed in, with machine still churning, slowly pour in the melted peanut butter and churn just until peanut butter is barely mixed in. **You want to leave "pools" and "streaks" of peanut butter.
Transfer to a freezer-safe container. Press a piece of plastic wrap right up against the custard and then cover tightly with a lid. Freeze for at least 2-3 hours before serving if you want it a little firmer.
Adjust the amount of peanut butter according to what you like. We love peanut butter so I always put closer to 3/4 cup in. It's a personal preference here.