Combine 1 1/4 cups milk with the cream in a saucepan over low-heat until warm. Turn off the heat, add chopped basil, and steep for one hour.
Strain out the basil, pushing to extract as much flavor as possible, then rewarm the mixture.
In a bowl, whisk together the sugar, cornstarch, and remaining cup of milk. Then, slowly whisk the warmed milk (from previous step) into the cornstarch mixture until combined. Return the mixture to the saucepan and cook until thickened, about 8 minutes.
Strain and stir in the vanilla paste.
Pour mixture into a container and freeze until firm, about two hours.