Bring a large pot of salted water to a boil over high heat. Cook 2 lobsters at a time until three-quarters done, approximately 10 minutes. Heat 2 large saute pans over high heat. Split the lobsters in half, brush with olive oil and season with salt and pepper to taste. Place the lobster in the pan, flesh-side down and cook until nicely browned. Turn over and cook shell side down for 1 more minute.;
Cava-Tarragon Butter Sauce:
Place the shallots, Cava and vinegar in a medium nonreactive saucepan. Boil the liquid over high heat until reduced to 2 tablespoons. Whisk in the heavy cream and cook for 1 minute. Reduce heat to medium and add the butter, piece by piece, whisking well after each addition until all has been used. Season with salt and white pepper and add the tarragon.;
Heat 2 tablespoons of oil in a medium saute pan over high heat until almost smoking and saute the tomatillos for 4 minutes. Place the tomatillos in a food processor and add the parsley, basil, tarragon, and water and puree.
Heat the garlic in a medium saucepan and saute until soft. Add the wine and cook until dry. Add the clam juice and reduce by 3/4. Add the lemon juice and season with salt and pepper, to taste. Add tomatillo-herb puree and cook for 2 minutes.
Heat olive oil and butter in a medium saucepan over medium heat. Add onions and garlic and cook until soft. Add rice and cook for 2 minutes. Add boiling water 1 cup at a time and cook until all liquid has been absorbed by the rice. Repeat until all water is used. Fold in the tomatillo-herb puree and cook until mixture has absorbed most of the liquid. Fold in the herbs and season with salt and pepper, to taste.