Saute coriander in oil and butter in large saucepan until fragrant. Add garlic and onion. Cover pot and soften vegetables over low heat for a few minutes. Add green tomatoes and saute a few minutes longer, until slightly softened. Add broth, herbs, salt and sugar. Cover and simmer about 45 minutes, until tomatoes are completely soft.
Process sauce in blender or food processor until smooth. Return to pan, heat to just below boiling, add cheese, and stir until cheese is elted and sauce is thick and smooth. Serve over spaghetti and garnish with fresh herb sprigs.