Purée the melon and 1/2 cup sugar in a food processor, scraping down the bowl, as needed, until smooth.

Scrape the mixture into a non-aluminum bowl and stir in the half-and-half. Taste and adjust if and as needed with the additional sugar; the mixture should taste a touch too sweet at room temperature if it is to taste perfect when frozen. Add the ginger, lemon, or lime juice by 1/2 teaspoons until the flavor rounds on your tongue. Don't be shy; splash in a bit more and watch the flavor grow.

Freeze in an ice cream maker according to the manufacturer's instructions.

Store with a piece of plastic wrap pressed directly on the surface.

Let soften slightly before serving.