Heat the cream, milk, honey and vanilla beans and seeds in a heavy saucepan over medium heat until hot, being careful not to let the mixture boil and curdle.
Lightly whisk the egg yolks in a medium heatproof bowl, then slowly drizzle 1 cup of the hot cream mixture into the yolks while whisking. Pour the yolk mixture into the saucepan of cream; heat, stirring constantly, until the custard thickens slightly and coats the back of a wooden spoon, again being careful not to let it boil and curdle.
Pour through a fine-mesh strainer to remove the vanilla beans and any bits of cooked egg yolk. Stir in the vanilla extract. Cover the custard with plastic wrap and refrigerate until cold, about 6 hours. You can speed this process dramatically by partially submerging the bowl of custard in a larger bowl of ice water to form an ice bath and stirring the custard occasionally until cold. The colder the custard is, the faster the machine will be able to freeze it for ice cream.
Follow the directions on your ice cream maker to freeze. Once the mixture is frozen, put it into containers and allow it to "ripen" for at least 2 hours in the freezer.