Boil the whole potatoes until they are soft (about 45 minutes). While still warm, peel and pass through vegetable mill onto clean pasta board.
Make well in center of potatoes and sprinkle all over with flour, using all the flour.
Place egg and salt in center of well and using a fork, stir into flour and potatoes, just like making normal pasta. Once egg is mixed in, bring dough together, kneading gently until a ball is formed. Knead gently another 4 minutes until ball is dry to touch.
Bring 6 quarts water to boil and add 2 tablespoons of salt.
Pinch off walnut-sized balls of gnocchi dough and roll with rolling pin to form 2-1/2 inch circle of dough.
Place one quarter of plum and a 1/2 teaspoon sugar on each and fold dough around plum pieces to form a package with no tears or rips, relatively smooth and evenly coated. Continue until all are finished. If making pasta in advance, cook and the shock in cold water.
Place gnocchi in water, 10 at a time, and carefully stir to prevent sticking to sides or bottom of pan. Allow to float and continue cooking for 2 minutes. Meanwhile, melt butter in a 12-inch saute pan over medium heat.
Add bread crumbs and then add cooked gnocchi and toss gently to coat.
