Using an egg topper or egg scissors, cut off wider end of each egg, or place each egg on its side on a towel, and use a serrated knife to saw halfway through wider end; remove yolk and white, and reserve whites and yolks of 2 eggs for ice cream (if using knife, saw through rest of shell after removing yolks and whites). Discard tops.
Bring a large pot of water to a boil. Rinse shells under hot running water; carefully pull out the membrane. Boil shells 1 minute. Break off any loose bits from edge, and make sure opening is large enough for a teaspoon. Drain upside down on a clean towel. When completely dry, turn right side up, place in mini muffin tins, and freeze.
Prepare an ice-water bath in a large bowl; set aside. In a small heavy-bottom saucepan, combine half-and-half, 1/3 cup sugar, and the salt. Bring to a simmer over medium-low heat, stirring occasionally until the sugar is dissolved. Remove from heat; stir in white chocolate until completely melted.
In the bowl of a mixer fitted with the whisk, beat reserved 2 egg whites and 2 egg yolks with remaining 1/3 cup sugar on medium-high until pale yellow, about 4 minutes.
With mixer on low, gradually pour in white-chocolate mixture; continue to beat until just combined. Beat in heavy cream. Return mixture to saucepan. Continue cooking over medium heat, stirring constantly with a wooden spoon, until mixture is thick enough to coat the back of the spoon and hold a line drawn with your finger, about 6 minutes. Remove from heat, and pour custard through a fine-mesh sieve into medium bowl; discard solids.
Set bowl of custard in ice-water bath; stir occasionally until cool, about 10 minutes. (At this point, the custard can be refrigerated up to 3 days before freezing; place plastic wrap directly on surface of custard first to prevent a skin from forming.) Place custard in an ice-cream maker, and freeze according to manufacturer's directions.
Transfer ice cream to a pastry bag with no tip. Remove eggshells from freezer. Working quickly, pipe ice cream into shells; return shells to freezer.
Just before serving, refrigerate ice-cream- filled eggs until slightly softened, about 10 minutes. Place eggs in nests; serve immediately.
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