Combine all of the ice cream ingredients in a bowl or large measuring cup. I often use a 2-quart measuring cup, to give me plenty of space for sloshy stirring while giving me that handy pour spout. Stir until the sugar and salt are melted and you don't feel any grainy bits left.

Refrigerate until fully chilled, then churn in your ice cream maker according to the manufacturer's instructions.

Put the butter, sugar, and salt in a heavy-bottomed pan. I use a saucier, but a sauce pot is fine. Heat on medium (stirring as needed) until the sugar melts and takes on a toast golden brown color (watch carefully so you don’t burn it!), then take it off the heat and add the Baileys Irish Cream. The sugar is probably going to seize and harden, but don't worry about that. Put it back on the heat and cook, stirring as needed, until the sugar melts and combines with the Baileys to make a thick caramel.

Add milk or cream as needed, until you have a slightly thickened sauce - this will thicken significantly as it cools, so you need it to be much thinner when it's hot. Otherwise, you'll end up with something more like an oozey caramel candy.

Serve the caramel warm (just barely warmed, not off-the-stove screaming hot) over the finished ice cream, or let it cool, then refrigerate until needed. If it thickens too much when cooled, you can reheat it and add more milk or cream until you have the consistency you're looking for.

Serve the ice cream with a drizzle of the caramel sauce, and pass extra at the table, if you like.