To make egg noodles, place flour, water, seasoned salt, 2 teaspoons olive oil, and 1 egg + 2 yolks in a mixing bowl. Using a stand mixer and dough hook, mix until combined and continue to knead for 8 minutes. (In lieu of stand mixer, knead by hand.) Cover and let rest for 10 minutes. Divide dough into four portions; roll each portion on a lightly floured surface to a 12 x 12-inch square. Let rest for 20 minutes. Cut squares into 1/4-inch wide strips and hang to dry on cooling racks. Dry for a day or two and break into 2- to 3-inch lengths.
In a stockpot saute carrots, red pepper, green onions and garlic in 2 tablespoons olive oil for 5 minutes. Add chicken broth, garlic and black pepper. Bring to boiling; add noodles. Reduce heat and cook for 5 minutes or until noodles are at desired cooked level. Add chicken. Beat remaining 2 egg whites and 4 eggs and pour in thin stream into soup. Cook for 2 minutes, then stir to break up egg. Serve immediately.