In a saucepan, whisk together the cream, milk, brown sugar, granulated sugar, cocoa powder, egg yolks, espresso powder and salt. The mixture will look lumpy, but it will smooth out as you cook it.
Over medium heat, warm the custard while whisking constantly, just until it begins to steam and is barely bubbling around the edges. Don't let it come to a boil.
Remove from the heat and whisk in the vanilla and the chopped chocolate until smooth. Pour the custard into a blender (or use an immersion blender) and blend for 30 seconds to ensure all the cocoa powder is broken down for a very smooth texture. Cover the bowl of custard, with plastic wrap resting right against the surface of the custard, and refrigerate until well chilled, preferably overnight.
Pour the chilled custard into your ice cream maker and churn according to the manufacturer's instructions, until it has a soft-serve consistency. During the last minute of churning, add the bourbon.
Scrape the ice cream into a lidded container and freeze until firm, at least 8-10 hours.
