Preheat the oven to 350 degrees F. Grease two cookie sheets.
Sift together the flour and the baking soda in a in a medium bowl and set aside. In a large mixing bowl, using a hand-held mixer, cream together the butter, brown sugar and granulated sugar. Beat in the instant pudding mix until incorporated.
Beat in the eggs, one at a time, and then add the flour mixture, chocolate chips and vanilla. Using a 1/4-cup-size scoop, drop onto the prepared cookie sheets, leaving a little space between the cookies. Flatten each cookie ball about halfway. Bake until the cookies are golden brown, 12 to 15 minutes. Remove from the oven and cool on a wire rack. Repeat with the remaining dough.
When cool, put 1 scoop (about 1/4 cup) of vanilla ice cream on a cookie and top with another cookie and press to form a sandwich. Repeat with the remaining cookies and ice cream. If not serving right away, place in the freezer on a tray. Dust with powdered sugar and serve.
This recipe has been updated and may differ from what was originally published or broadcast.