In medium saucepan, combine broth, water, broccoli, and seasonings. Bring to boiling over high heat. Stir in noodles. Reduce heat to medium and cook, stirring occasionally, until noodles are tender, 10 to 12 minutes. Stir in chicken. Keep hot over low heat. In large saucepan, bring 3 inches of water to boiling. Break eggs into small bowl. Reduce heat to simmering. Hold bowl close to water?s surface and let eggs slide separately into water. Cook until whites are set and yolks begin to thicken but are not hard, 3 to 5 minutes. Pour 1 cup of the soup into each of 4 bowls. Lift eggs with slotted spoon and gently lower onto soup. Sprinkle each with 1 tablespoon of the cheese.