Make the Dough: Adjust oven rack to middle position and preheat to 350°F (180°C). Combine butter, coconut oil, sugar, salt, baking powder, and vanilla in the bowl of a stand mixer fitted with a paddle attachment. Mix on low to moisten, then increase to medium and beat until light and fluffy, about 5 minutes, pausing to scrape bowl with a flexible spatula halfway through. With mixer running, add egg and continue beating until smooth. Reduce speed to low, add flour, and mix to form a stiff dough, then add frozen cookie pieces and continue mixing until well combined.