Fill a large pot with 2 quarts water, add 2 teaspoons salt and bring to a boil over high heat.
Remove from the heat; add the shrimp, and let sit (off the heat) until the shrimp is pink-orange and slightly firm, 1 to 2 minutes.
Transfer to a colander with a slotted spoon and rinse under cold water until cool enough to handle.
Return the water to a boil, then reduce to a simmer over low heat. Gently lower the eggs into the water with a large ladle, two at a time, and let simmer exactly 14 minutes.
Meanwhile, peel and chop the shrimp and put them in a large bowl.
Saute the bacon in a skillet over medium-high heat until firm and just turning golden brown, 4 to 5 minutes.
Drain on paper towels.
When the eggs are done, transfer to a colander and rinse under cold water until cool enough to handle, about 2 minutes. Peel the eggs and halve lengthwise. Separate the whites from the yolks; coarsely chop the whites and add them to the bowl with the shrimp. Press the yolks through a mesh strainer into a separate bowl.
Add the mayonnaise, hot sauce, mustard and 1/2 teaspoon salt to the yolks and whisk until the mixture is the consistency of cake batter, about 1 minute.
Using a rubber spatula, fold the yolk mixture into the shrimp mixture until the shrimp is evenly coated with the deviled-egg dressing. Season with salt and pepper and toss again. (The shrimp and deviled-egg salad will keep in the refrigerator, covered, for 2 days.)
To serve, load 1 piece of lettuce and 1 tomato slice into each of the bun halves; spread about 1/2 cup of the shrimp and deviled-egg salad in each.
