Preheat oven to 350ºF and line a baking sheet with parchment paper. Whisk together the flour, baking soda, and salt until combined; set aside.
In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma.
Once browned and nutty, turn off the heat and pour butter into a shallow dish. Refrigerate until solid.
In the bowl of a stand mixer fitted with the paddle attachment, cream the solidified browned butter, cream cheese, and brown sugar together until lightened in color and creamy.
Beat in the egg and vanilla, followed by the butterscotch pudding mix. Scrape down the sides of the bowl, then dump in the flour mixture. Beat until just combined. Stir in toffee bits and butterscotch chips by hand.
Using a cookie scoop (or two spoons), scoop dough onto lined baking sheet about 2 inches apart. Slightly press the tops of each cookie to allow for even baking. Gently press in a few M&Ms into the top of each cookie. Bake for 9-11 minutes, or until edges begin to turn golden. Centers may appear undercooked, but will set up as they cool.
Allow cookies to cool on baking sheet for 3-4 minutes before transferring to a wire rack to finish cooling.