Preheat oven to 350 degree F. Line a 9x13-inch baking pan with aluminum foil and spritz with vegetable-oil cooking spray.
Combine flour, hot-cocoa powder, and salt in a medium bowl. Beat the butter and sugar in a medium bowl with an electric mixer on high until light and fluffy, about 3 minutes. Add egg and beat well. Gradually add the flour mixture and mix until just combined, scraping down sides of bowl.
Spread the dough evenly in the prepared pan. Bake until firm to the touch, 30 to 35 minutes. Transfer to a wire rack to cool for 5 minutes. Lift foil and shortbread cookie from pan and place it on a cutting board. Slice in half lengthwise. Cut each half crosswise into 3/4-inch pieces. After cookies have cooled, dot them with frosting. Let dry before storing.