In a small saucepan, bring milk to a simmer over medium heat. Add green tea and let steep for 1 hour. Strain.

Whisk together eggs and sugar in a large bowl. Add cream to infused milk and bring just to a simmer. Remove from heat and slowly add 2 tablespoons of warm milk mixture to egg mixture, whisking constantly. Slowly pour in remaining milk while whisking vigorously. Transfer to saucepan and return to medium-low heat. Cook slowly, being careful not to boil and stirring constantly, until thick enough to coat the back of a spoon, about 8 to 12 minutes. Transfer to a bowl and chill, uncovered, until cold.

Freeze in an ice cream maker according to instructions. Transfer to an airtight container and freeze for 3 hours or overnight. Store for up to one week in freezer.