Start by preheating your oven to 350°F (175°C). Grease and flour a 9x13-inch baking pan, or line it with parchment paper for easy removal later.

In a medium saucepan, combine 1 cup of Dr Pepper, 1/2 cup of butter, and 1/4 cup of cocoa powder. Cook this mixture over medium heat, stirring constantly until the butter melts and everything is well combined. Once smooth, remove the saucepan from the heat and let the mixture cool slightly.

In a large mixing bowl, whisk together 1 cup of granulated sugar, 1 cup of packed brown sugar, 2 cups of all-purpose flour, 1 teaspoon of baking soda, and 1/2 teaspoon of salt. Pour in the cooled Dr Pepper mixture and stir until just combined.

Add 1/2 cup of buttermilk, 2 large eggs, and 1 teaspoon of vanilla extract to the bowl. Mix until the batter is smooth and all ingredients are well combined. Avoid overmixing, as it can make the cake tough.

Pour the batter into the prepared pan, spreading it evenly. Bake in the preheated oven for 25-30 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.

While the cake cools, prepare the frosting. In a medium saucepan, melt 1/2 cup of butter over medium heat. Add 1/4 cup of cocoa powder and 1/4 cup of Dr Pepper, stirring until smooth. Remove the mixture from heat.

Gradually whisk in 3 cups of powdered sugar until the frosting is smooth and creamy. Lastly, stir in 1 teaspoon of vanilla extract.

Once the cake is completely cool, spread the frosting evenly over the top. Allow the frosting to set for a few minutes before slicing and serving.