Add the egg yolks to a medium sized bowl, use a fork to mix them until smooth. Set aside.
In a saucepan over medium-low heat add the brown sugar, salt, heavy cream, and whole milk. Stir until the mixture is smooth. Bring to a simmer, stirring occasionally, just until bubbles form on the sides of the pan. Remove from heat.
Temper the egg yolks: Scoop out ½ cup of the hot cream mixture and gradually pour it over the egg yolks, while stirring constantly. Now, slowly drizzle the egg yolks back into the saucepan, stirring constantly.
Place the saucepan over low heat. Heat until thickened, should coat the back of a spoon, about 5-7 minutes, if you have an instant read thermometer it should read 170°F. Do not bring the mixture to a boil.
Remove from heat and stir in the vanilla extract.
Set a fine mesh sieve over a large bowl. Pour the butterscotch ice cream mixture through the sieve. Cover the bowl and chill in the refrigerator for 4 hours or up to 1 day.
Once chilled, remove from the fridge and pour the mixture into an ice cream maker and churn according to manufacturers directions. Once finished churning serve immediately with butterscotch sauce drizzled overtop! Or if you want the ice cream to be a harder and more firm consistency, place it in an airtight container, cover, and freeze until ready to serve.