In a non-stick pan, heat 1 tablespoon of butter over high heat until it foams and subsides. Crack 1 egg into the pan and cook, sunny side up, seasoning with salt and pepper and removing to a plate as it is finished cooking. Repeat this process with the remaining 3 eggs and butter.
Meanwhile, in a large bowl, combine the radicchio, prosciutto, walnuts, oil, vinegar, and salt and pepper, to taste, and toss well to coat. Divide the salad evenly among 4 plates and top each serving with a sunny side up egg and a hard boiled, finely chopped egg.