Preheat oven to 400 degrees. Put egg whites and sugar in the bowl of an electric mixer fitted with the paddle attachment; mix 30 seconds. Mix in flour, cocoa, and salt. Mix in butter, cream, and vanilla; mix 30 seconds.
Bake cookies: Using plastic lids or heavy card stock, cut out 1 circle (2 3/4 inches in diameter) and 1 half circle (4 1/2 inches long). Using the outlines as stencils, spread dough on a Silpat baking mat with an offset spatula to make 4 circles and 2 half circles. Transfer mat to a baking sheet; bake cookies until starting to set, 6 to 7 minutes.
Shape cones: Remove half circles with an offset spatula while still hot. With straight sides of cookies facing up, quickly bring edges together to form cones, holding until set. Repeat three more times with remaining dough.
Make sandwitches: Dip cones in melted chocolate to coat 1/4 inch of rims. Place cones on 8 whole cookies; press 1 candy onto band of each hat. Freeze until set, 10 to 20 minutes. Brush remaining 8 whole cookies on one side with melted chocolate. Freeze until set, 10 to 20 minutes. Cut ice cream using technique described above. Place on coated cookies; top with hats.
For neat rounds of ice cream, snip away the carton with scissors, cut ice cream into 3/4-inch-thick slices, and make shapes with a 2 1/2-inch cookie cutter. Freeze leftovers in an airtight container.