In a 1- to 2-quart pan over medium heat, stir brown sugar and butter until butter is melted, sugar is dissolved, and mixture is bubbly, 3 to 4 minutes. Whisk in ½ cup whipping cream until smooth; remove butterscotch mixture from heat. Add vanilla and bourbon, if using.
In a 3- to 4-quart pan over medium-high heat, combine remaining 1 cup whipping cream and the half-and-half; bring to a simmer.
Meanwhile, in a bowl, beat egg yolks to blend. Whisk ½ cup of the warm cream mixture into egg yolks, then pour egg yolk mixture into pan with cream. Stir constantly over low heat just until mixture is slightly thickened, 2 to 4 minutes. Immediately remove from heat.
Whisk in butterscotch mixture, then when well-combined (most of butterscotch should be dissolved into custard), pour through a fine strainer into a clean bowl.
Cool in an ice bath, then cover and chill at least 2 hours, or overnight (up to 1 day).
Freeze mixture in an ice cream maker according to manufacturer’s instructions. Serve softly frozen, or transfer ice cream to an airtight container and freeze until firm, at least 6 hours or up to 1 week.