For best results, prepare sandwiches the day before serving.
Combine flour and cocoa and set aside. Melt together chocolates and butter in a medium bowl set over simmering water.Stir with a wooden spoon until smooth. Add salt and sugar, stirring until sugar is almost completely dissolved, about 5 minutes. Remove bowl from heat. Beat in eggs one at a time, then fold in flour mixture until well combined. Let dough cool slightly in bowl; it will be very soft. Divide into quarters and spoon onto plastic wrap. Flatten each piece of dough with a wooden spoon, wrap well, and refrigerate for at least 2 hours or overnight.
When dough is firm, cut each piece in half. Cover a work surface with plastic wrap. Place piece of dough on the plastic and cover with another sheet; roll out to a thickness of 1/8 inch. Keeping dough in plastic, transfer to a baking sheet; place in freezer. Repeat process until all dough is rolled out. Stack sheets of dough in freezer and chill until very firm, about 30 minutes.
Heat oven to 350 degrees. Line a baking sheet with parchment paper. Remove sheets of dough from freezer one at a time. Place on a parchment-lined work surface and peel off plastic wrap. Using a 2 1/2-inch round cookie cutter, cut out dough and transfer circles to baking sheet. Work quickly to prevent the dough from becoming too soft.
Bake until cookies are firm to the touch, 14 to 16 minutes, turning baking sheet halfway through baking. Remove from oven and let cookies cool slightly on pan. Transfer to a wire rack to cool completely.
While cookies are baking, transfer ice cream (working with one flavor at a time) to the bowl of an electric mixer fitted with a paddle attachment. Place ice cream in refrigerator to soften slightly, about 15 minutes. Beat ice cream on medium-high speed until soft enough to spread, about 2 minutes, or beat with a wooden spoon. Set bowl in a second bowl filled with ice. Spoon some softened ice cream onto a cookie and top with another cookie. Place sandwich in freezer and continue process. After the ice cream has hardened, wrap sandwiches tightly in plastic wrap. Chill overnight in freezer.