In a pitcher or large measuring cup, stir together the whipping cream, half-and-half, sugar, and vanilla extract until sugar has dissolved.
Pour about 1/2 cup of mixture into a pint-size plastic bag and seal carefully, squeezing out extra air. Place each sealed bag into a second pint-size bag, again squeezing out extra air. Seal carefully.
Fill each gallon-size plastic bag about halfway with ice and add 1/2 cup coarse salt. Place one sealed small bag into the large bag, squeeze out most of the air, and seal the large bag.
Wear mittens or thick gloves, or wrap the bag in a towel to protect hands against the extreme cold. Shake and massage the bag for about 5 minutes or until mixture thickens into ice cream. Add more salt and ice to the outer bag if ice cream hasn't formed after 10 minutes of continuous motion.
Remove the outer pint-size bag before you open the inner bag so you don't get any of the salty ice on your ice cream!