Beat together half-and-half, white sugar, and brown sugar in a large bowl with an electric mixer on medium speed until the sugars have dissolved.
Stir heavy cream and vanilla extract into the mixture until smooth.
Pour the mixture into an ice cream maker and freeze according to manufacturer’s instructions.
About 2 minutes before the end of freezing time, pour Irish cream liqueur into the ice cream maker; allow the machine to finish freezing the ice cream.
Transfer ice cream into a freezer-proof container with a tight lid and place in freezer until ice cream is hardened, at least 4 hours.
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