In a medium pan, melt the chocolate and 4 tbsp butter over medium-low heat, stirring until smooth. Let cool for 10 minutes.
In a medium bowl, whisk together the flour, cocoa, baking powder and salt.
Using an electric mixer, beat the granulated sugar, brown sugar, eggs and vanilla on medium speed until smooth, about 3 minutes. Mix in chocolate mixture on low speed. Mix in flour mixture until just incorporated. Transfer the dough to a bowl; cover and refrigerate for 1 hour or overnight.
Position racks in the upper and lower thirds of the oven; preheat to 325 degrees . Line 2 cookie sheets with parchment. Roll the dough into 16 balls about 1 1/2 inches in diameter. Place 8 balls on each sheet, flatten slightly and bake, switching pans halfway through baking, until puffed and cracked on top, about 15 minutes. Let the cookies cool on the pans for 5 minutes, then transfer to a rack and let cool completely.
Using a handheld electric mixer, beat the remaining 2 tbsp butter with the confectioners' sugar until blended. Gradually mix in the half-and-half on medium speed, then beat at high speed until fluffy. Set aside.
Let the ice cream soften until spreadable. Working on a parchment-lined cookie sheet, spread a heaping tbsp of frosting onto the flat side of half the cookies. Top the flat side of the remaining cookies with 1/4 cup softened ice cream. Sandwich the cookies together and freeze until firm, about 1 hour. Serve, or wrap and freeze for up to 2 weeks.