Combine first 4 ingredients in a small saucepan; bring to a boil. Cover and remove from heat; let stand 15 minutes. Strain cranberry mixture through a fine mesh sieve into an 11 x 7-inch baking dish; discard solids. Cool to room temperature; stir in lime juice. Cover and freeze for about 45 minutes. Stir cranberry mixture every 45 minutes until completely frozen (about 3 hours). Remove mixture from freezer; scrape entire mixture with a fork until fluffy.
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WOW!!!! This is a blow-your-socks-off kind of dish. I would consider it more of a cross between a palate-cleanser and a dessert but you can definitely serve it at the end of a meal even though it's not too sweet. I did use unsweetened pure cranberry juice instead of cranberry cocktail which may have changed my results but I highly recommend it, and did not seed the jalapeno and this came out as one of the most flavorful and interesting dishes anyone I served it too has ever tried. It also makes 3/4 cup servings not 1/2 or you could serve 1/2 a cup and have enough for 6 or seven people. I think it could be a really interesting addition to a Thanksgiving menu although certainly not traditional. Try it!
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