To make snapper, preheat oven to 400°F. Rinse fish and pat dry. Place several lemon circles and fennel slices in cavity of fish.
In two large mixing bowls, whisk together 4 egg whites and 1 cup water in each, until frothy, about 1 minute. Divide salt between the bowls. Work with your hands to mix thoroughly.
Select a baking dish 2 to 3 inches longer than the fish. Fill it with 1/3 of the salt mixture; place fish on top and cover completely in remaining salt.
Bake 40 minutes; an instant-read thermometer inserted through layer of salt and into flesh of fish should read 135°F.
Bring fish to table in baking dish, with a large plate for salt shell and a mallet or hammer. Crack salt shell and break away from fish. Peel off snapper skin with a fork. Divide fish into fourths, and serve with vinaigrette.