In a large pan over LOW heat, melt butter. Add onion and cover with lid.
Cook for 4 minutes. Add broth and cook for 5 minutes. Cover and set aside.
In a small pan over LOW heat, add half and half, thyme and lobster base. Cook for 6 minutes to slightly thicken the sauce. Set aside to cool.
Blend both mixtures together in a food processor. Remove and move to bowls and add lobster meat to each. Serve immediately.