Preheat oven to 400 degrees F (200 degrees C). Spray 2 baking sheets with cooking spray.
Melt bittersweet chocolate in top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Remove from heat and cool slightly.
Beat egg whites in a large bowl using an electric mixer until soft peaks form; gradually beat in 1 cup confectioners' sugar until mixture resembles soft marshmallow creme.
Whisk 1/2 cup confectioners' sugar, cocoa powder, cornstarch, and salt together in a bowl. Beat confectioners' sugar mixture into egg white mixture on low speed until a soft meringue forms. Stir in cooled melted bittersweet chocolate and dark chocolate until dough becomes stiff.
Place remaining 1/2 cup confectioners' sugar in a bowl. Roll 1 rounded tablespoon of dough into a ball and roll ball in the confectioners' sugar, generously coated all sides. Repeat with remaining dough and sugar. Arrange cookies 2 inches apart on the baking sheets and sprinkle coarse salt over each.
Bake in the preheated oven until tops start to crack, 9 to 10 minutes. Cool cookies on the baking sheets for 10 minutes before transferring to a wire rack to cool completely.