In a large kettle or Dutch oven combine 4 cups of the chicken broth, the leeks, mushrooms, fines herbes, and pepper. Bring to boiling. Reduce heat and simmer, covered, for 10 to 15 minutes or until leeks are tender. Cool slightly.
Place one-third of the leek mixture in a blender container or food processor bowl. Cover and blend or process until smooth. Repeat with remaining leek mixture. Return all of the smooth leek mixture to the kettle or Dutch oven; stir in 1 cup of the remaining chicken broth.
Combine the remaining 1 cup of the chicken broth and the flour until smooth. Stir into hot broth mixture along with shredded cheese and parsley. Cook and stir until slightly thickened and bubbly and cheese melts. Stir in whipping cream; heat through.