Combine sweetened condensed milk, evaporated milk, whole milk, peanut butter, sugar, and vanilla extract in an ice cream maker, and stir until sugar dissolves.

Freeze according to manufacturer's directions until it reaches soft-serve consistency, adding chopped peanut butter cups halfway through the freezing cycle.

Transfer ice cream to a one- or two-quart lidded plastic airtight container. Cover surface with plastic wrap and seal.