Fit a stand mixer with the paddle attachment (or use a large mixing bowl and electric mixer). Add the egg, softened unsalted butter, packed light brown sugar, granulated sugar, and vanilla. Beat on medium-high until the mixture is creamed and well combined, about 4 minutes.
Stop the mixer and scrape down the bowl sides. Add the rolled oats, all-purpose flour, baking soda, and salt. Beat on low speed just until combined, about 1 minute.
Stop the mixer, scrape the bowl, then add 1 cup M&Ms and the semi-sweet chocolate chips. Mix on low until evenly distributed, about 15–30 seconds.
Use a large cookie scoop, 1/4-cup measuring cup, or your hands to form about 12 equal mounds of dough. Roll each mound into a ball and flatten slightly. If desired, press a few extra M&Ms onto the tops for appearance.
Place the dough mounds on a large plate or tray, cover tightly with plastic wrap, and refrigerate for at least 3 hours (up to 5 days). Do not bake unchilled dough to avoid thin, flat cookies.
Preheat the oven to 350°F (177°C). Line a baking sheet with a Silpat or spray it with cooking spray. Arrange chilled dough mounds on the sheet at least 2 inches apart (about 8 per sheet).
Bake for 11–12 minutes, until the edges are set and the tops are just set (they may still look slightly pale and glossy). Do not overbake; cookies will firm up as they cool.
Allow the cookies to cool on the baking sheet for about 10 minutes before serving.