Cut off roots and trim leaf ends of leeks cut each leek lengthwise in half and separate leaves. Rinse well with cold running water to remove any sand. Cut leeks crosswise into 3⁄4 inch pieces.
Cut root end and stalks from fennel bulb discard. Cut the fennel bulb lengthwise into thin wedges.
In 5 quart Dutch oven or saucepot, heat 1 tablespoon olive oil over medium high heat until hot. Add 1 tablespoon butter melt. Add chicken and cook until chicken is golden and just loses its pink color through out. With slotted spoon, transfer chicken to medium bowl.
To drippings in Dutch oven, add mushrooms and cook, stirring often, until golden (do not overbrown). Transfer mushrooms to bowl with chicken.
To Dutch oven, add remaining 1 tablespoon olive oil heat until hot. Add remaining 1 tablespoon butter or margarine melt. Add carrots, leeks, fennel, potatoes, bay leaf, and tarragon. Cook vegetables 10 to 15 minutes, until fennel is translucent and leeks are wilted, stirring occasionally.
Add wine cook 2 minutes, stirring. Add chicken broth and water; heat to boiling over high heat. Reduce heat to low cover and simmer 20 minutes or until vegetables are tender.
In cup, mix half and half and flour until smooth. Stir half and half mixture into vegetable mixture heat to boiling over high heat. Reduce heat to medium cook 1 minute to thicken slightly. Stir in chicken, mushrooms, peas, and salt heat through. Discard bay leaf.