In a large saucepan set over medium heat, heat the half-and-half and vanilla bean until the milk steams.
Remove the pan from the heat, cover it, and let the vanilla bean steep for 15 minutes. Remove the bean and scrape the vanilla seeds from the pod, adding them to the hot half-and-half.
In a large bowl, beat the sugar and egg yolks with a whisk until well-blended. Add the hot half-and-half in a stream, whisking constantly. Return the egg mixture back to the pan and whisk in the salt and green tea powder. Heat over medium heat, stirring constantly with a wooden spoon or heatproof spatula.
When the mixture begins to thicken, after about 3 minutes, remove it from the heat and transfer it to a heatproof bowl. Add the cream and milk, and stir to cool the custard. Cover and refrigerate overnight.
Process the cold custard in an ice cream maker following the manufacturer's directions. When frozen (it will be the consistency of soft-serve ice cream), transfer the ice cream to a container and freeze for 2 hours to firm up before serving.
NOTE: Matcha tea powder can be found in Asian grocers or online.