Combine heavy cream and milk in a large bowl. Whisk together egg, egg yolks and cornstarch in a medium bowl.
In a large pot set over medium-low heat stir together brown sugar, water and salt until sugar dissolves. Increase heat to medium-high and allow to boil without stirring, until the mixture turns thick and dark amber in color, occasionally scraping the sides of the pot with a rubber spatula, about 7 minutes.
Immediately add in cream mixture and whisk to combine and dissolve sugary bits. Bring mixture to a boil, watching pot carefully to avoid boiling over, then reduce heat to medium.
Whisk half of this hot caramel mixture into the egg mixture. Return mixture to pot and whisk to combine until custard boils and thickens, about 2 minutes. Remove from heat and stir in butter and bourbon.