Heat 1 cup half-and-half, heavy cream, and milk in a saucepan over medium heat until begins to simmer, about 10 minutes; stir in tea mix and bring to a boil. Reduce heat to low; simmer 5 to 10 minutes.
Beat egg in a bowl; whisk in about ¼ cup warm cream mixture, 1 teaspoon at a time. Whisk egg mixture back into cream mixture; cook until custard slightly thickened, about 1 minute. Strain custard through a cheesecloth-lined sieve into a bowl; discard tea leaves.
Beat remaining 1 cup half-and-half, sweetened condensed milk, and vanilla extract into custard until well blended; cover the bowl with plastic wrap and chill in the refrigerator, 6 hours to overnight.
Pour mixture into an ice cream maker and freeze according to manufacturer’s instructions, mixing in ¾ crushed cookies when semi-solid. Fold in remaining ¼ cookies when ice cream reaches desired consistency. Serve immediately or transfer to an airtight container and freeze until firm.