Prepare for baking: Preheat your oven to 350°F (175°C) and make sure to grease and flour a 9x13-inch baking dish so that your cake won't stick.
Mix the batter: In a large bowl, combine yellow cake mix, whole milk, sweet potato puree, large eggs, vegetable oil, and vanilla extract. Mix on low speed just until everything is combined.
Make the filling: In a separate small bowl, whisk together dark brown sugar, ground cinnamon, ground nutmeg, and a pinch of salt.
Layer the cake: Pour half of the cake batter into your prepared dish. Sprinkle the brown sugar filling evenly over the top, and then pour the remaining batter over the filling.
Bake the cake: Place the dish in the oven and bake for 35 to 40 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Make the icing: While your cake is cooling, whisk together powdered sugar, whole milk, and vanilla extract until it is nice and smooth.
Drizzle and serve: Once the cake has cooled slightly, drizzle the delicious vanilla icing all over the top. Serve it warm and enjoy every single bite.