Stir together bourbon, honey, vanilla, 1 cup of the cream, and 1/2 cup of the milk in a medium bowl. Set aside.
Whisk together egg yolk, sugar, salt, and remaining 1 1/4 cups heavy cream and 3/4 cup milk in a medium saucepan over medium. Bring to a simmer, and cook, stirring constantly and scraping the bottom of the pan, until mixture thickens slightly and lightly coats the back of a spoon (170°F to 175°F), about 12 minutes. Pour custard through a wire-mesh strainer into bowl with bourbon mixture.
Chill bowl over an ice bath, and then cover and refrigerate 8 hours. Transfer mixture to the freezer container of a 1-quart electric ice-cream maker, and freeze according to the manufacturer's instructions. Transfer to a freezer-safe container, and fold in chopped Bourbon Ball Candies. Freeze overnight.