Return the shrimp water to a boil, then reduce heat to low and gently add eggs. Simmer eggs for 14 minutes. Meanwhile, peel and chop the cooked shrimp and place in a large bowl. Transfer eggs to a strainer; rinse under cold water until cool.
Cook bacon in a skillet over medium-high heat until golden brown. Transfer to a small bowl and crumble. Peel eggs; cut in half lengthwise. Separate the whites from the yolks; coarsely chop the egg whites and add to the bowl with the chopped shrimp.
Press the yolks through a mesh strainer and into a large bowl. Add mayonnaise, Tabasco, mustard, and 1 tsp. salt; whisk until mixture is consistency of cake batter, about 1 minute. Fold yolk mixture into shrimp mixture, add bacon and green onion; toss. Season with salt and pepper; toss.
Use 1 cup per sandwich, along with sliced tomatoes and lettuce. Serve immediately.