Heat the oven to 350°F. Line a 9-by-13-inch baking pan with parchment paper, leaving a 2-inch overhang on two sides. In a large bowl, whisk together the flour, salt, baking powder, and baking soda.
In a medium saucepan, heat the butter over medium-high heat until melted. Continue to cook the butter, swirling the pan often, until the milk solids turn golden brown and the butter smells nutty, about 5 minutes.
Remove the pot from the heat and let it cool for a few minutes. (This will prevent the egg from scrambling when you stir it in.) Stir in the brown sugar until smooth. Stir in the egg and the vanilla extract. Stir in the flour mixture. The batter will be thick.
Transfer the mixture into the prepared pan. Use a spoon or an offset spatula to smooth the dough to an even thickness. Sprinkle with the Heath bar pieces and press them lightly to adhere. Bake until the dough is puffed and set, 8 to 12 minutes. (To keep them chewy, be sure not to overbake.) Let pan cool on a rack. Using the parchment, lift the bar out of the pan and onto a cutting board. Cut into 12 pieces. (Or, cut the cookies into 24 rectangles to make smaller ice cream sandwiches.)
Turn 6 of the blondie cookies over on a work surface. Press ½ cup of ice cream onto each half and top with another cookie. Wrap each sandwich in plastic wrap and freeze until firm, at least 8 hours.
* Total Time includes chilling
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