Bring a large pot of water to boil, turn off heat, then add the lobster. Cover and cook for 3 minutes.
Remove all lobster meat from shell and cut into large chunks.
In a sauce pan, bring 1 tbsp water to simmer over medium-low heat. Whisk in butter one piece at a time, letting each piece melt before adding the next. (Make sure the butter does NOT come to a boil, otherwise the butter will separate.)
Add the lobster pieces, shrimp, basil, garlic, and tomatoes and cook on medium-high for 5 minutes, stirring occasionally.