If you want the bellinis to be frozen, either pre-freeze the puree or use frozen sliced peaches to lessen the need for ice. This will keep the drinks from getting watery. Traditional bellinis call for sparkling wine but that's hard to run through a blender without making a frothy mess. So I use the peach schnapps and some vanilla vodka with the frozen peach puree and a minimal amount of ice and then just add a "topping" shot glass of the champagne/prosecco/sparkling wine after you pour it into the glass.
Oh, and take a cue from the beer drinkers and put your glasses in the freezer to frost. You can add a sugar rim of Demerara if you want.
Here is a link to the recipe for the original belini created at Harry's bar in Venice Italy
http://www.gourmetsleuth.com/Recipes/Cocktails-Alcohol-402/Bellini-Cocktail-221.aspx
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Oh, and take a cue from the beer drinkers and put your glasses in the freezer to frost. You can add a sugar rim of Demerara if you want.
http://www.gourmetsleuth.com/Recipes/Cocktails-Alcohol-402/Bellini-Cocktail-221.aspx