Whisk matcha powder in a bowl to remove any lumps; add a splash of milk and whisk until matcha powder is completely dissolved. Gradually whisk remaining milk into matcha mixture.

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Combine cream and matcha mixture in a pot over medium-low heat; cook, stirring occasionally, until heated through, about 5 minutes.

Whisk sugar and eggs together in a bowl. Pour 1/2 cup hot matcha mixture into egg mixture; mix thoroughly. Repeat with remaining matcha mixture. Pour mixture back into the pot.

Cook and stir matcha mixture over medium-low heat until heated through, about 3 minutes. Remove from heat and cool to room temperature. Refrigerate until chilled, at least 4 hours.

Pour cooled matcha mixture into an ice cream maker and freeze according to the manufacturer’s instructions.