Prepare the Dry Ingredients - In a medium bowl, whisk together the flour and baking soda, and set aside.
Brown the Butter - Cut the butter into 14 even-sized pieces. In a medium skillet, melt 10 tablespoons of butter over medium-high heat. Continue cooking, swirling pan constantly until golden brown and a nutty aroma forms, 1 to 3 minutes. Remove the skillet from heat. Stir the remaining 4 tablespoons of butter into warm butter until completely melted.
Mix the Butter and Sugars – In a stand mixer fitted with the whisk attachment, add the browned butter to the mixing bowl. Add the granulated sugar, brown sugar, salt, and vanilla. Whisk on medium speed (setting 5) until combined, 30 seconds.
Whisk and Rest – Add the whole egg and yolk to the mixing bowl. Whisk on medium-high speed (setting 8) until smooth with no sugar lumps remaining, about 30 seconds. Let the mixture stand for 3 minutes.Repeat the whisking and resting process 3 more times: 30 seconds mixing at medium-high speed and 3 minutes of rest. After four total times, the mixture should be thick, smooth, and shiny.
Add the Flour and M&M's - Using a rubber spatula or wooden spoon, stir in the flour mixture until just combined, about 1 minute. Stir in ¾ cup M&M candies and chocolate chips. Give the dough a final stir to ensure no flour pockets remain.
Chill the Dough - Transfer the mixing bowl to the refrigerator for 30 minutes or the freezer for 15 minutes. Chill until the dough firms up, around 65 to 70ºF (18 to 21ºC). The dough should not feel overly sticky.
Preheat the Oven - Adjust the oven rack to the middle position. Heat to 375°F (191ºC). Line two baking sheets with parchment paper, then set aside.
Roll Out Dough Balls – Divide the dough into 16 portions, each about 3 tablespoons (about 60 grams, 2 ounces), and roll into a ball. If the dough starts to warm and get sticky, place it on a sheet tray and refrigerate for 15 minutes.For a rustic appearance, tear the dough balls in half, then press the two halves together again with the lumpy torn sides facing upward to expose the chips. Shape the sides of the dough back into a circle. If desired, use the remaining ¼ cup of M&M candies and press some on top, so they pop out of the surface.Arrange dough balls at least 2" apart on lined baking sheets, 8 portions per tray. Smaller baking sheets can be used but will require baking in 3 batches. The dough should be room temperature or just below for even cooking, around 65 to 70ºF (18 to 21ºC).
Bake the Cookies - Bake one tray at a time in the center position for 5 minutes, then rotate the baking sheet. Continue to bake until the edges are golden brown and the centers look slightly underdone, about 5 to 7 minutes. The color will continue to darken during cooling.
Let Them Cool - Leave the cookies on the baking sheet to cool for 3 to 5 minutes, then transfer them to a wire rack. Make sure the oven returns to 375°F (191ºC) before baking the remaining tray.
Notes
Make ahead: Cookie dough can be shaped into balls and refrigerated in a resealable plastic bag for up to 1 week or frozen for up to 2 months before baking. Break the dough apart and combine right before baking for a rustic surface.
Storing: Baked cookies can be stored in an airtight container at room temperature for 5 days or 3 months in the freezer.