Melt the butter in a 2 quart sauce pot. Sweat the zucchini and the salt to nearly au sec. It will look like a mushy paste. Take care not to burn as the mixture reduces.
Transfer to a blender with the spices and whole milk. Blend until smooth. Return to the pan along with the heavy cream. Bring to just below a simmer.
Whisk the egg yolks with the sugar until pale in color. Add 1/2 cup of the hot zucchini mix, whisking rapidly to incorporate. Add the yolk mixture back into the pot, whisking constantly.
Cook over med-low heat, stirring frequently, until mixture coats the back of a spoon. Strain the ice cream base through a fine-meshed sieve into a stainless steel bowl and chill over an ice bath. Stir in the beer and vanilla. Churn in an ice cream maker.