Bring 2 1/2 quarts of water to a boil in a large pot. Add 3 1/2 teaspoons salt.
In a medium bowl, combine the flour, and eggs. Add 2/3 cup water and remaining 1/2 teaspoon salt. Using your hand, mix to form a dough. Divide the dough into quarters.
Place one quarter of the dough on a small, dampened cutting board, flattening to 1/2-inch thick. Dip a paring knife into the boiling water, and cut the dough into 1-by-1/2-inch pieces directly into the boiling water. Repeat process with remaining dough. If the knife begins to stick to the dough, dip it into the boiling water.
As dumplings are cooking, stir water occasionally to prevent sticking. When dumplings rise to the surface, use a slotted spoon to transfer them to a colander to drain. Place in a medium and drizzle with remaining tablespoon oil, and toss to coat. Serve immediately.