Combine 3/4 cup heavy cream, 3/4 cup half-and-half, and mint leaves in a medium saucepan. Bring mixture just to a simmer, then cover and remove from heat. Let sit 30 minutes. Strain and discard mint leaves.
In a large bowl, whisk egg yolks and sugar. Whisk in the mint infused cream mixture. Pour mixture into the same saucepan you used to infuse the cream.
Cook over medium-low heat, stirring continuously, until mixture thickens and coats the back of a wooden spoon. This can take 4 to 10 minutes. If there are any curdled pieces of egg, pour mixture through a fine-meshed sieve. Otherwise pour into a bowl that is set in an ice bath.
Whisk in remaining cream and half-and-half as well as the vanilla, bourbon, and salt.
Refrigerate at least 2 hours and up to 24 hours.
Freeze in an ice cream maker according to manufacturer's instructions