Heat the heavy cream and milk in a medium saucepan over medium heat until small bubbles form around the edges. Remove from heat and add fresh mint leaves, pressing them gently with a spoon to release oils. Cover and steep for 30 minutes. Strain out the mint leaves, pressing them to extract maximum flavor.

Whisk egg yolks and sugar in a large bowl until pale yellow and slightly thickened, about 3 minutes. Slowly pour half the warm mint-infused cream into the yolk mixture while whisking constantly to prevent scrambling. Return the mixture to the saucepan and add the remaining cream.

Cook over medium-low heat, stirring constantly with a wooden spoon, until the mixture coats the back of the spoon and reaches 170°F on a candy thermometer. Remove from heat immediately and strain through a fine-mesh sieve.

Stir in bourbon, vanilla extract, simple syrup, and salt while the custard is still warm. Press plastic wrap directly onto the surface and refrigerate for at least 4 hours or overnight until completely cold and thickened.

Pour the chilled custard into your ice cream maker and churn according to manufacturer's instructions, typically 20-25 minutes, until it reaches soft-serve consistency. Transfer to an airtight container and freeze for at least 2 hours for scoopable consistency.