Preheat an oven to 350 degrees F. Combine the flour, baking soda and salt in a large bowl. Beat the butter, granulated sugar and brown sugar together until light and fluffy (about 3 minutes) on medium speed. Beat in the eggs, one at a time, then add the vanilla.
At a low speed, beat in the dry ingredients in batches, until just blended, do not over beat or the dough will get tough. Stir in the chocolate chunks and chopped chocolate using a wooden spoon until well combined. Wrap the bowl with plastic and chill dough 30 minutes or overnight.
Shape or scoop into 2-inch balls using a spoon or ice cream scoop. Place on cookie sheets, about 3 inches apart. (About 6 cookies per tray with 2 trays per batch.) Flatten them slightly and bake until golden brown around the edges, about 15 to 18 minutes. Let cool on tray about 3 minutes. Transfer the cookies to a cooling rack and cool completely.
Remove the ice cream from the freezer about 5 minutes to thaw/soften slightly before scooping. Using a 2-ounce ice cream scoop, place 3 scoops of ice cream (about 1/2 cup) onto the flat side of a cookie. Top with another cookie and press gently to flatten slightly. Using an offset spatula or a butter knife, smooth the ice cream around the edges of the cookie until the ice cream is flush with the outside of the cookie. Serve immediately or re-freeze until ready to serve. If re-freezing, thaw the sandwich for about 4 to 5 minutes before serving.