Whisk together the cream, milk, sugar, salt, and vanilla. Pour the mixture into your ice cream maker and freeze according to the manufacturer's directions. While the ice cream is churning, make the hot fudge sauce and let it cool on the counter.
When the ice cream has finished churning, scoop ⅓ of the ice cream into a freezer-safe container, drizzle generously with about ⅓ of the cooled hot fudge sauce, and sprinkle with peanut butter cups.
Repeat these layers twice with the remaining ice cream, sauce, fudge, and peanut butter cups. Freeze until firm. Serve with additional hot fudge sauce and extra peanut butter cups, if desired.