Heat the milk and cream. Heat the milk, cream, ½ cup of sugar in a saucepan over medium heat, until warm and steaming, or about 170℉.
Beat the eggs and remaining sugar. Meanwhile, beat the egg yolks and remaining ¼ cup of sugar until light yellow in color and well combined. They will noticeably change color when mixed enough. (A hand mixer on low will work well.)
Temper the eggs by adding half of the warm milk mixture, a half cup at a time. Then add the egg mixture back into the saucepan with the rest of the milk and cream.
Heat the custard. Heat the saucepan, which now contains all of the milk, cream, sugar, and eggs, over medium heat, stirring constantly, until it reaches 185℉. The custard is ready when it has thickened enough to coat the back of a spoon. Do not boil.
Strain the custard and add the vanilla and salt. Hold a cheesecloth or a strainer over a bowl to strain the custard. Scrape the vanilla beans and add them and the vanilla extract along with the salt and stir to combine. Immediately place the custard into the refrigerator to chill. Chill in bowl for 4-24 hours, until very cold.
Churn. Remove custard from fridge and set up ice cream maker, according to manufacturer's instructions. Churn until it is the consistency of soft serve ice cream. Make sure everything is cold before you churn. The custard will take hours to cool in the fridge, or you can cool it quickly by transferring it into a bowl that has been set over an ice bath and stir it frequently until the temperature drops.
Freeze. Freeze the chilled custard in your ice cream maker according to the manufacturer's instructions. When it has reached the consistency of soft serve ice cream, place it into a loaf pan, cover with plastic wrap, and freeze until firm. Keep it covered in the freezer for up to a week.