In a large pot over low heat, melt 2 tablespoons of the butter. Add the onion, stir, and cook until soft, about 5 minutes. Add the potatoes and 6 cups broth or water, and bring to a simmer. Season with salt and pepper. Cook for about 30 minutes, until potatoes begin to soften.
Meanwhile, in a large skillet over medium-high heat, melt 2 tablespoons of the butter. Add about a third of the lobster and sauté for about a minute or so. Set aside. Repeat with the remaining lobster and butter.
Add lobster to onion and potato mixture; stir over medium heat. Add canned milk and basil, taste, and adjust seasoning with additional salt and pepper. If mixture is too thick, add milk or cream. Serve.