For the lobster salad and sandwich:
2 – 1.25 -pound live lobsters or 1 3/4 cups cooked lobster meat
4 ounces great-quality thick-sliced smoked bacon (3 to 4 slices)
1 stalk celery, finely diced
1 tablespoon fresh lemon juice
Sea salt and freshly ground black pepper
4 slices toasted brioche bread, grilled hot dog bun, or crusty roll
4 Romaine or Buttercrunch lettuce leaves
2 to 4 thick slices vine-ripened tomato
For the basil aioli:
1 cup mayonnaise
1/4 cup thinly sliced basil leaves
2 1/2 teaspoons olive oil
Salt and freshly ground black pepper to taste
To cook the lobsters: Fill a large pot with enough water to come about 2 inches up the sides of the pot. Bring to a boil over high heat. Add a pinch of salt. Add the lobsters, shell side down, cover, and let steam for 12 minutes. Drain the lobsters and let cool.
When the lobster is cool enough to handle, remove the meat from the tails and claws and very coarsely chop. You should have between 1 2/3 and 1 3/4 cups of meat. Set aside in a medium bowl. Save the bodies and shells for snacking on or for making a lobster broth).
Cook the bacon in a skillet until crisp on both sides, about 10 minutes. Drain on paper towels and set aside.
To make the basil aioli: In a small bowl, whisk together the mayonnaise, basil, and oil. Season to taste with salt and pepper. Cover and refrigerate; the aioli will keep for several days. (You will have more aioli than you need here; save for use in chicken sandwiches, chicken salad, roast beef and cheddar cheese sandwiches, and more.)
Add the celery to the lobster. Fold in 2 tablespoons of the aioli and the lemon juice to lightly coat the ingredients. Season with salt and pepper. Store in refrigerator for up to 6 hours if not making the BLTs at once.
To assemble a BLT: Spread bread with a light dollop of the basil aioli, just under a tablespoon on each piece of bread. Divide bacon, lettuce leaves, tomato slices, and lobster salad between 2 slices of the toast. Cover with the remaining 2 slices toast. Slice the sandwiches in half on the diagonal.