Heat milk and heavy cream in a saucepan until small bubbles appear, then remove from heat.

Whisk egg yolks, sugar, and salt for 3 minutes.

Temper eggs by slowly whisking in the warm milk mixture.

Return the mixture to the saucepan and cook over low heat until it thickens and coats the back of a spoon (about 10 minutes).

Whisk matcha with hot water until dissolved, then whisk it into the custard.

Pour the mixture through a sieve into a large bowl, cover with a towel, and cool to room temperature.

If using an ice cream maker, freeze according to manufacturer's instructions.

If not using an ice cream maker, pour the mixture into an airtight container and freeze until set (about 3 hours).

Remove from the container and cut into 3-inch pieces, process in a food processor until smooth.

Return to the airtight container and freeze again.

Repeat the freezing and chopping process until smooth.