In a medium bowl, whisk together peanut butter and sugar until sugar is completely dissolved, about 1–2 minutes. Add cream and vanilla extract and stir until mixed. Add milk and salt, and continue stirring until it is completely incorporated.
Prepare your ice cream maker, turn it on, and pour the ice cream into the machine. Churn according to your machine’s directions (usually 20–25 minutes). About 15 minutes in, pour the toffee pieces in and continue churning until you achieve a thick, soft serve consistency.
Scrape into a freezer-safe container and freeze for at least 4 hours, ideally overnight. Ice cream is better fresh, so consume within two weeks of making it, or it will become dense.
When you are ready to serve, heat the chocolate chips and coconut oil in the microwave until thin and liquid. Let cool slightly.
Pour slowly over the scooped ice cream, garnish with some big flake sea salt, and serve immediately. If the shell does not harden immediately, place the sundae in the freezer for about 30 seconds.