Stir milk, cream, and brown sugar in a medium saucepan over medium-high heat until mixture comes to a rolling boil (it may look curdled, but this is normal).

Meanwhile, whisk yolks, sugar, and salt in a medium bowl until mixture is pale yellow and sugar dissolves.

Place a medium metal bowl in a large bowl of ice water; set a fine-mesh sieve over metal bowl.

Whisking yolk mixture constantly, gradually add hot cream mixture (to prevent the eggs in the yolk mixture from scrambling). Return to pan.

Stir custard over low heat until thick enough to coat the back of a spoon, about 2 minutes.

Immediately strain custard into metal bowl.

Let custard cool completely, stirring often.

Stir 1 Tbsp. bourbon and vanilla into custard. Add more bourbon to taste by teaspoonfuls, if desired.

DO AHEAD Custard can be made 1 day ahead. Cover and chill. Process custard in an ice cream maker according to manufacturer's instructions.