In a large mixing bowl stir together the 2 cups flour and the salt. Make a well in the center of the dry mixture. In a small mixing bowl combine egg whites, water, and oil. Add to the dry mixture; mix well.
Sprinkle kneading surface with 1 tablespoon of the remaining flour. Turn dough out onto floured surface. Knead until dough is smooth and elastic (8 to 10 minutes total), adding 1 to 2 tablespoons additional flour as needed. Cover and let the dough rest for 10 minutes.
Divide dough into 4 equal portions. On a lightly floured surface, roll each portion into a 12-inch square (about 1/16-inch thick). Let stand, uncovered, about 20 minutes. Loosely roll up dough into a spiral; cut into 1/4-inch-wide strips. Shake the strands to separate and cut into 2- to 3-inch lengths. If using a pasta machine, pass each portion through machine according to manufacturer's directions until dough is 1/16 inch thick. Cut into 1/4-inch-wide strips. Cut strips into 2- to 3-inch lengths.
To store cut noodles, spread them on a wire cooling rack. Let noodles dry overnight or until completely dry; place in an airtight container and refrigerate for up to 3 days. Or, dry the noodles for at least 1 hour and seal in a freezer bag or freezer container; freeze for up to 8 months.
Cook homemade noodles for 1-1/2 to 2 minutes or until tender but still firm, allowing a few more minutes for dried or frozen noodles. Drain noodles well.