In a small sauce pan, add the unsalted butter. Heat on the stove over medium-low heat. Allow the butter to melt and start to bubble a bit. Swirl butter around or gently stir with a wooden spoon. Continue cooking the butter until the milk solids begin to brown, about 10 minutes. Turn off the butter and remove from heat. Allow brown butter to cool.
Preheat oven to 350°F. Line 2 baking sheets with a silicone baking mat or parchment paper. Set aside.
In a mixer, combine the cooled brown butter, brown sugar, and granulated sugar. Mix for about 30 seconds until smooth. Add in the eggs and vanilla extract. Mix until everything is combined and mixture is smooth.
Add in the flour, baking soda, and salt. Mix until all the flour is incorporated. Add in the M&Ms and white chocolate chips and give it another stir.
Scoop dough with a medium cookie scoop. Roll scooped dough into balls and place on your prepared baking sheets a few inches apart.
Bake for 11-12 minutes. Remove from oven and top with more M&Ms and white chocolate chips, if desired. Enjoy warm or cool completely!