1. Heat oven to 350 degrees F.
2. Sift flour, cocoa powder, baking soda and salt into medium-size bowl.
3. In large bowl, beat butter with an electric mixer on medium-high speed until creamy, about 1 minute. Add both kinds of sugar; beat until mixture is creamed, about 2 minutes. Beat in the eggs, one at a time, beating well after each addition. Beat in vanilla until well blended.
4. On low speed, beat in flour mixture until smooth. Stir in chocolate pieces.
5. For each cookie, drop level 1/4 cup of the dough onto an ungreased baking sheet, spacing cookies 2 inches apart.
6. Bake in 350 degrees F oven until slightly puffed and dry on tops, 13 or 14 minutes. Cool cookies on sheets on wire racks for 1 minute. Move cookies to rack; cool completely.Drizzle:
7. Select 12 cookies to be sandwich tops. In small bowl, whisk together confectioners sugar and water. Drizzle over tops. Let dry 15 minutes.To sandwich:
8. Slice the paper container off of ice cream; slice ice cream into eight 1/2-inch-thick slices. Working quickly, place one slice onto a sheet of waxed paper. Using a 3-inch cookie cutter, cut out 1 circle of ice cream. Transfer circle to a waxed-paper-lined baking sheet. Gather up ice-cream scraps and press into cookie cutter to form a second 3-inch circle. Transfer this circle to baking sheet. Repeat with remaining ice-cream slices, transferring baking sheet to freezer when 12 circles are formed (save excess ice cream for snacking). Freeze circles 1 hour.
9. Sandwich each ice-cream circle with 2 cookies; return cookie sandwiches to freezer for an additional 2 hours. Cookies can be stored in an airtight container in freezer for up to 2 weeks.