Cream the goat cheese and cream cheese. Season with salt and olive oil. Refrigerate until ready to use.
Bring a pot of water to a boil. Drop the eggs into the water and cook for 10 minutes. Drain and cool the eggs. Peel off the shells, set aside one egg for garnish, and chop the other eggs. Mix the chopped egg with the rest of the ingredients and refrigerate until ready to use.
Mix all of the ingredients and set aside until ready to use
Trim off all sides of the pumpernickel, making it an even rectangle. Slice the loaf into three pieces lengthwise. Begin building the sandwich by spreading a thin layer of the icing on all three of the bread slices. Top one with salmon and then cucumbers. Place another bread slice on top with the icing toward the cucumber. Spread the egg salad on the second slice and top with more cucumber. Place the final slice of bread on top with the icing toward the cucumber. Give the sandwich a gentle press.
Ice the sandwich like a normal cake starting with a crumb coat. Place the sandwich on a serving platter. Garnish the cake with radish sprouts, sliced tomato, shaved radish, egg slices, shaved red onion, and dill. Refrigerate until ready to serve.