Warm the honey in a small heavy-bottomed saucepan over medium heat. Cook, without stirring, for 3 to 4 minutes, until it darkens to a deep amber and smells toasty. Remove from the heat immediately.
Warm the cream and milk in a separate saucepan until small bubbles appear at the edge. Slowly whisk the burnt honey into the warm cream (it will bubble up — be careful), then keep warm over low heat.
Whisk the egg yolks, sugar, and salt in a bowl until pale. Slowly ladle the warm cream into the yolks, whisking constantly, then return the mixture to the saucepan.
Cook over medium-low heat, stirring with a wooden spoon, until the custard thickens enough to coat the back of the spoon. Strain through a fine-mesh sieve, stir in the vanilla, and chill overnight.
Churn in an ice cream maker according to the manufacturer's instructions. Transfer to a container and freeze for at least 2 hours to firm up.
