1. Butter and flour 12 muffin cups, each about 2 1/2 inches wide.
2. In a 1- to 2-quart pan over medium-low heat, stir 1/4 cup butter and the unsweetened chocolate often until melted and smooth, 5 to 6 minutes. Remove from heat and stir in sugar, vanilla, and egg. Add 1/2 cup flour and mix until smooth. Spoon batter equally into muffin cups (about 3 1/2 teaspoons per cup).
3. Bake in a 350° regular or convection oven just until brownie edges spring back when lightly pressed, 6 to 8 minutes. Loosen brownies from pans with a knife and tip out onto a rack. Let cool completely, about 20 minutes.
4. Line unwashed muffin cups with 6-inch squares of plastic wrap, letting edges extend above cup rims. Return brownies to cups.
5. Scoop a 1/4-cup portion of ice cream onto each brownie. (If ice cream is too hard to scoop, warm--1 carton at a time, if using more than 1--in a microwave oven at half power (50%) just until slightly softened, checking every 10 seconds.) With the bottom of a glass, press ice cream in each cup down to form a firm, even layer. Wrap pans airtight and freeze until ice cream is firm, at least 3 hours.
6. Lift brownie stacks in plastic from muffin cups; pull off plastic wrap and discard. Place stacks slightly apart on a 12- by 15-inch baking sheet and put back in freezer.
7. Pour 1 inch of water into the bottom of a double boiler or into a 3- to 4-quart pan; bring to a boil over high heat, then reduce heat to a simmer. Combine chocolate chips and shortening in top of double boiler or a metal bowl or pan that fits snugly in pan with water; set over simmering water (bottom of bowl shouldn't touch water). Stir mixture occasionally just until chocolate is soft, 3 to 5 minutes. Remove from double boiler and stir until chocolate is melted and mixture is smooth. If hotter than 95° to 100°, continue stirring often until cooled to this range. Replace simmering water in bottom of double boiler or in other pan with very hot tap water. Set chocolate mixture over hot water (top container should sit on, or just above, water), not over heat.
8. Working quickly with 2 forks, lower 1 brownie stack at a time into chocolate and turn to coat all sides. Lift stack out and return, flat side down, to chilled pan (don't move stack once it touches frozen pan, or chocolate will stick to pan and peel off ice cream; if this happens, patch with melted chocolate). Reserve extra chocolate mixture for other uses. Freeze stacks until ice cream is firm, at least 30 minutes.
9. Dab any ice cream drips off stacks with a paper towel. Serve stacks, or wrap individually airtight and freeze up to 2 weeks.
Nutritional analysis per sandwich.