Preheat the oven to 350°F. Grease a 9×13-inch baking dish.
Prepare one 15.25-ounce box of white cake mix according to the package directions. Pour the batter into the prepared baking dish and bake according to the package directions, until a toothpick inserted into the center comes out clean.
While the cake is baking, prepare the cinnamon filling. In a small bowl, stir together 1/2 cup melted butter, 1/2 cup packed brown sugar, 1/2 cup sweetened condensed milk, and 1 tablespoon ground cinnamon until smooth and well combined.
Remove the cake from the oven. While the cake is still warm, use the handle of a wooden spoon to poke holes evenly across the top of the cake.
Slowly pour the cinnamon filling over the warm cake, spreading it gently so it seeps into the holes.
Let the cake cool completely to room temperature.
To make the frosting, beat 8 ounces softened cream cheese and 1/4 cup softened butter until smooth and creamy.
Add 2 cups powdered sugar, 1 tablespoon milk, and 1/2 teaspoon vanilla extract. Beat until smooth and well combined. Add 1 to 2 additional tablespoons of milk, as needed, until the frosting reaches a smooth, spreadable consistency.
Spread the frosting evenly over the cooled cake.
Serve at room temperature or refrigerate until chilled.
Cover and refrigerate any leftovers.