Make the shortbread layer: Preheat the oven to 350˚. Line a 9-by-13-inch baking pan with foil, leaving a 2-inch overhang on the two short sides; lightly coat with cooking spray. Arrange the shortbread cookies in a single layer in the prepared pan.
Make the cookie-dough layer: Whisk the flour, baking soda and salt in a medium bowl. Beat the butter and brown sugar in a large bowl with a mixer on medium-high speed until pale and fluffy, 3 to 5 minutes. Beat in the egg and vanilla until smooth. Reduce the speed to low and beat in the flour mixture until just combined. Stir in the baking chips and M&M's.
Make the brownie layer: Melt the butter and chocolate in a medium saucepan over medium heat, stirring occasionally. Let cool slightly. Stir in the granulated sugar and eggs until smooth and glossy. Stir in the flour, cocoa powder and salt until no lumps remain.
Spoon about half the brownie batter over the shortbread cookies and spread gently to cover. Flatten handfuls of the cookie dough mixture into small patties and lay on top of the brownie batter in a single layer. Spread the remaining brownie batter on top to cover.
Bake until a toothpick inserted into the center of the brownies comes out clean, 55 to 60 minutes. Transfer to a rack and let cool 1 hour in the pan. Carefully remove from the pan using the foil overhang; transfer the brownies to the rack to cool completely. Cut into pieces.