Heat the milk and cream. Heat the milk, 1¼ cup of the cream, and all the honey in a saucepan over medium heat until warm and steaming, or about 170℉. You'll have just a small amount of cream left to beat with the eggs.
Beat the eggs. Meanwhile, beat the egg yolks and remaining cream 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. Add half of the warm milk mixture, a half cup at a time, to the eggs. 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, honey, 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. Take off the heat when it reaches temperature.
Strain and add vanilla. Hold a cheesecloth or a strainer over a bowl to strain the custard. Add the vanilla bean and vanilla extract along with the salt and stir to combine. Immediately place the custard in the fridge to chill. Chill for 4-24 hours, until very cold.
Churn and freeze. Churn the cold custard in the ice cream maker, according to the manufacturer's instructions. When it has reached the consistency of soft serve ice cream, place into a loaf pan, cover with plastic wrap, and freeze until firm, at least 3 hours.