Preheat oven to 350 degrees F (175 degrees C).
In a mixer bowl outfitted with paddle attachment, beat together the butter, shortening, brown sugar, and sugar on low until bound together, then on medium-high until smooth and fluffy. Add vanilla, continue beating, then add one egg. Beat until combined, then add the other egg and beat until fluffy. Separate into two equal batches and set aside.
In a large bowl, stir or whisk together flour, baking soda, baking powder, salt, and pudding mix. Separate into two equal batches. Whisk 1/4 cup cocoa into one of the two batches.
Mix one batch of wet ingredients with one batch of the dry ingredients. In another bowl, mix the remaining wet and dry batches. Place half of the chocolate chips into one batch of dough and the remaining chocolate chips into the other batch of dough.
Use a cookie scoop to measure out a scoop of regular dough and a scoop of chocolate dough and stack them together, one on top of the other and place on insulated baking sheets.
Bake in a preheated oven for 8-11 minutes, until light golden brown. Remove from oven; allow cookies to cool 5 minutes on baking sheets before removing them to wire racks to cool completely.