For the dill sour cream: In a medium bowl, whisk to combine the sour cream, dill and some salt and pepper. Cover and refrigerate until ready to serve.
For the hash brown cake: Heat the butter and oil in a large nonstick skillet over medium-high heat. In a large bowl, stir to combine the potatoes, flour, paprika, salt, black pepper, cayenne pepper, eggs, garlic and onion.
Spread the potato mixture in an even layer to cover the bottom of the skillet and use a spatula to press gently. Cover and cook until the potatoes are warmed through and crisp on the bottom, 8 to 10 minutes. Uncover the skillet and remove from the heat. Place a platter larger than the skillet directly over the top of the skillet and invert the potato cake onto the plate.
Cut into wedges and serve with the dill sour cream and sprigs of dill for garnish.