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1
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.
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2
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).
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3
Cook the bacon in a skillet until crisp on both sides, about 10 minutes. Drain on paper towels and set aside.
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4
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, chickensalad, roast beef and cheddar cheese sandwiches, and more.)
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5
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 the refrigerator for up to 6 hours if not making the BLTs at once.
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6
To assemble a BLT: Lightly toast the bread. Spread the toast with a light dollop of the basil aioli, just under a tablespoon on each piece of bread. Divide the 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.