Preheat oven to 350°F. Spray a 9x13-inch baking pan generously with cooking spray, including corners and sides.
In a large mixing bowl, combine yellow cake mix, sour cream, vegetable oil, eggs, and vanilla extract. Mix with a stand mixer, hand mixer, or whisk for about 2 minutes until smooth. The batter will be very thick.
Pour half of the batter into the prepared pan. Use a spatula to spread evenly across the bottom.
In a small bowl, mix together brown sugar, cinnamon, and nutmeg until well combined.
Sprinkle the entire cinnamon mixture evenly over the batter in the pan.
Carefully spoon the remaining batter over the cinnamon layer. Gently spread to cover without disturbing the cinnamon sugar too much.
Using a butter knife, make 3-4 gentle swirls through just the top layer. Don't over-swirl or go all the way to the bottom.
Bake for 30-35 minutes, or until a toothpick inserted in the center comes out with just a few moist crumbs. The top should be golden and spring back when lightly pressed.
While the cake bakes, prepare the glaze by whisking together confectioners' sugar, milk, and vanilla in a medium bowl until completely smooth with no lumps.
Remove cake from oven and let cool for 10 minutes only. Pour glaze over the warm cake and spread evenly with a spatula.
Let the cake sit for at least 20 minutes to allow the glaze to set before cutting into 24 squares and serving.