Preheat oven to 325°F (163°C). Grease a 9x13-inch baking pan or line it with parchment paper.
Mix cake batter: In a large bowl, combine cake mix, sour cream, eggs, and vegetable oil. Mix until well combined and smooth.
Make cinnamon swirl: In a small bowl, mix brown sugar and cinnamon together.
Layer the cake: Pour half of the batter into the prepared baking pan. Sprinkle the cinnamon-sugar mixture evenly over the batter. Add the remaining batter on top and gently swirl with a knife or spatula (do not overmix).
Bake for 38–42 minutes, or until a toothpick inserted in the center comes out clean.
Prepare glaze: While the cake bakes, whisk together powdered sugar, milk, and vanilla extract until smooth.
Glaze the cake: Once the cake is done baking and still warm, pour the glaze evenly over the top.
Cool slightly, then slice and serve warm or at room temperature.