Preheat the oven to 350 degrees F (175 degrees C). Grease and lightly flour a 9x13-inch baking pan.
Combine brown sugar, white sugar, flour, cinnamon, and salt for crumb topping in a medium bowl and toss lightly with a fork.
Grate the butter into the flour mixture using the largest side of a box grater. Combine with a fork or pastry knife until ingredients are combined and have the appearance of very coarse meal; the pieces should be loose and not sticking together. Stir in pecans and set topping aside.
Combine flour, baking powder, and salt for cake in a large bowl.
Combine butter, white sugar, and brown sugar in the bowl of an electric stand mixer. Beat on medium speed until light and fluffy, about 2 minutes. Add sour cream, eggs, and vanilla extract and mix, scraping the bowl as needed, for 1 minute. Add flour mixture on low speed in thirds, alternating with milk, ending with the flour mixture. Mix until just smooth. Stir in cranberries with a spoon.
Spread batter into the prepared baking pan and sprinkle topping evenly over the batter.
Bake in the preheated oven until a toothpick inserted into the center of the cake comes out clean, about 55 minutes. Cool in the pan on a wire rack for about 10 minutes. Serve warm.
