In a medium saucepan, combine the whole milk, brown sugar, and white sugar.
Warm the mixture over medium heat, stirring until the sugars fully dissolve.
In a small bowl, mix the cornstarch with 2 tablespoons of cold milk to make a slurry.
Stir the cornstarch slurry into the warm milk mixture.
Bring the mixture to a gentle simmer and stir for 2–3 minutes until slightly thickened.
Remove the pan from heat and stir in the heavy cream and vanilla extract.
Transfer the mixture to a bowl and refrigerate for at least 4 hours, or until completely cold.
While the base chills, make the brown sugar syrup by combining 1 cup brown sugar and 1/2 cup water in a saucepan.
Bring the syrup to a boil over medium heat, then simmer for 5 minutes until slightly thickened.
Remove from heat and let cool completely.
To cook the boba pearls, bring 5 cups of water to a boil in a separate saucepan.
Add 1 cup of quick-cooking tapioca pearls to the boiling water.
Cook the pearls for 5 minutes, stirring gently.
Turn off the heat, cover the pot, and let the pearls sit for another 5 minutes.
Drain the boba and rinse under cold water.
Transfer the cooked pearls to a small bowl and cover with 3 tablespoons of the brown sugar syrup. Let them soak until ready to use.
Once the ice cream base is fully chilled, pour it into your ice cream maker and churn according to the manufacturer’s instructions (usually 20–30 minutes).
In the last 5 minutes of churning, gently fold in the soaked boba pearls.
Swirl in an extra few spoonfuls of brown sugar syrup if desired.
Transfer the finished ice cream to a freezer-safe container and freeze for at least 2 hours to firm up before serving.
Scoop and serve in a bowl, cone, or cup, and drizzle with more brown sugar syrup if desired.