Make the gnocchi dough. In a large saucepan, boil the potatoes until they are soft, about 45 minutes. While still warm, peel and pass through a vegetable mill or ricer onto a clean pasta board. Place the chopped peppers in a towel and wring to extract as much liquid as possible.
Make a well in the center of the potatoes and sprinkle all over with the flour, using all the flour. Place the egg, salt, and chopped pepper in the center of the well and, using a fork, stir into the flour and potatoes, just like making pasta. Once the egg and roasted peppers are mixed in, bring the dough together, kneading gently until a ball is formed. Knead gently another 4 minutes until the ball is dry to the touch.
Bring 6 quarts of water to a boil and add about 2 tablespoons salt.
Heat the olive oil in a 12 to 14-inch saute pan over medium heat until almost smoking. Add the garlic and cook until light brown, about 30 seconds. Add the olives and tomatoes and cook for 3 minutes. Keep warm.
Divide the dough into 4 balls. Flour the board and roll each ball into a 1-inch-thick rope. Cut the rope into 1-inch-long pieces and let the pieces dry slightly before shaping. Roll the pieces down the tines of a fork to establish the classic shape. Repeat with the remaining gnocchi dough. Drop the gnocchi, 10 to 15 at a time, into boiling water and cook until they float to the surface.
Gently remove with a slotted spoon and add to the pan with the olives and tomato. Add the marjoram. Toss until coated over medium heat, about 1 minute. Place in a warm bowl and serve.