Mix the Coffee and Milk: In a medium saucepan, combine heavy cream, whole milk, and sugar. Stir constantly over medium heat until the sugar dissolves and the mixture is heated, but not boiling. Add instant coffee and stir until fully dissolved. Remove from heat.
Add the Coffee and Vanilla: Stir in the cooled brewed coffee and vanilla extract. Let the mixture cool to room temperature, then refrigerate for about an hour to chill.
Churn the Mixture: Once chilled, pour the mixture into an ice cream maker and churn according to the manufacturer's instructions (usually for 20–25 minutes) until it reaches a soft, creamy consistency.
Freeze to Set: Transfer the ice cream to an airtight container and place in the freezer for at least 4 hours, or until fully set.
Serve and Enjoy: Once set, scoop and enjoy your homemade cappuccino ice cream!