Melt butter in a large saucepan. Add onions and saute over medium heat until tender-add garlic and saute one minute more (do not burn!). Sprinkle flour over all and cook. stirring constantly 3 minutes.
Gradually add broth and water, stirring constantly. Bring to boil. Add broccoli, thyme & pepper. Cover and simmer until broccoli is tender, approx 5 minutes.
Stir together cream and egg yolk until blended. Stir a little of the soup mixture into the egg yolk and stir until blended. Add back to saucepan, stirring to blend. Add cheese and cook stirring constantly until smooth and blended. Do not boil.