Preheat the oven to 350 degrees F (180 degrees C).
In a mixer, combine the Unsalted Butter (1 cup), Vegetable Shortening (1 cup), Brown Sugar (1 1/2 cup), Granulated Sugar (1/2 cup), Egg (1) and egg yolk, and Vanilla Extract (2 1/2 teaspoon).
In another bowl, mix together the All-Purpose Flour (2 1/2 cup), Salt (1 teaspoon), Baking Soda (1 teaspoon), and Baking Powder (1 teaspoon).
Slowly add the dry ingredients to the wet ingredients and mix well. Add the Semi-Sweet Chocolate Chips (1 package) and continue to mix until incorporated.
Roll the dough into golf ball-sized balls. Place dough balls on an ungreased cookie sheet.
Bake two sheets at a time for 10-12 minutes, or just until the edges are lightly browned. Rotate the cookie sheets half way through the baking time.
Transfer cookies to a cooling rack and allow to cool completely before assembling the sandwich.
Place a scoop of Ice Cream (to taste) onto the flat side of the cookie, then top with the second cookie. Gently press to allow the ice cream to spread to the edges.
Immediately roll the cookie sandwich in the chocolate chip mini morsels and place on a plate in the freezer. Repeat with the remaining cookies. Cover the dish with plastic wrap and freeze for at least 4 hours.