In a medium saucepan, heat 1 cup heavy cream and milk over medium-heat. Stirring constantly, let mixture reach just the point of simmer and remove from heat.
While cream and milk mixture is heating, whisk egg yolks and sugar together in a separate bowl until pale and thick. Slowly pour about half of the warm milk mixture into the bowl with the yolk mixture, whisking continuously (this will temper the mixture, meaning the egg yolks will warm without scrambling). Then pour egg/milk mixture back into the saucepan with the rest of the mixture.
Continue to cook over low heat, whisking all the while, about 5 minutes. When the mixture coats the back of the spoon or reaches 170-180 degrees on a thermometer, remove saucepan from heat. Stir in the peanut butter until it's melted into the mixture.
Pour the remaining cup heavy cream into a large bowl. Strain the egg/milk mixture into the cream, stirring all the while. Stir in the vanilla extract and salt. Cover bowl and refrigerate for at least 4 hours.
Process the ice cream in your ice cream maker according to manufacturer’s instructions. When it’s just about done, mix in peanut butter swirl mixture and chopped Reese's cups.
Put ice cream in airtight containers and put back into the freezer for at least 4 hours, but overnight if possible.