In a heavy 4-qt. saucepan, bring turtle meat and water to a boil. Skim off foam. Chop 1 onion and set aside. Quarter the other onion; add to saucepan along with bay leaf, cayenne pepper and salt. Cover and simmer for 2 hours or until the meat is tender.
Remove meat with a slotted spoon and cut into 1/2-in. cubes; set aside. Strain broth and set aside. Rinse and dry saucepan; melt butter over medium-high heat. Cook chopped onion until tender.
Add flour; cook and stir until bubbly and lightly browned.
Whisk in reserved broth; cook and stir until thickened. Reduce heat; stir in tomato puree and Worcestershire sauce.
Simmer, uncovered, for 10 minutes.
Add chicken broth, eggs, lemon juice and meat. Simmer for 5 minutes or until heated through.