Preheat the oven to 350°F. Line large baking sheets with parchment paper. In a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, beat the butter, sugars, and vanilla together on medium speed until light and fluffy, about 3 minutes. Beat in the eggs one at a time, stopping to scrape down the bowl between each egg, until well incorporated and fluffy.
Add the pudding mix, flour, baking soda, and salt. Beat on low speed until combined. Scrape down the bowl. Add the chocolate chips and stir in by hand until evenly distributed. Using a 2-tablespoon scoop, scoop the cookie dough and place on the lined baking sheet 2 inches apart. Bake one sheet at a time for 10 to 12 minutes or until the edges are lightly golden but the tops are still a little shiny. Let cool for a few minutes on the pan, then transfer to a wire rack and cool completely. Store the cookies in an airtight container for up to 1 week.