Beat the eggs well in a large bowl. In a heavy saute pan, about 8 inches in diameter, melt 2 teaspoons of butter over extremely low heat. Add the beaten eggs, and cook over the lowest possible heat for 45 minutes. Stir occasionally to prevent a skin from forming on the surface. The eggs will thicken very slowly. When they begin to thicken, stir gently with a large spoon to create large flaps in the scrambled eggs. The eggs will be darker in color and thicker than normal scrambled eggs.

Spread 1 teaspoon of butter on each of the bread halves. For each smorrebrod, at 2 corners that face each other diagonally, place 2 slices of cucumber. At the other 2 corners, place about 1/4 teaspoon of salmon roe. Cover the rest of the bread with about 1/12 of the egg mixture. Repeat until all smorrebrod are done. Garnish with sprigs of fresh dill.