Preheat the oven to 375 degrees F.
Heat a large saucepan over medium-high heat; add the anchovies. Cook the anchovies, stirring, until beginning to soften, about 1 minute. Stir in the butter, garlic and 1/4 cup parsley. Cook the anchovy mixture until the butter begins to brown, about 2 minutes. Remove from the heat and stir in the lemon juice. Add the stale bread and toss until coated with the anchovy mixture. Transfer the bread to a large rimmed baking sheet and bake until golden brown, about 10 minutes. Set aside to cool completely.
Meanwhile, combine the mayonnaise, tomato paste, tarragon, 1/4 cup water and the remaining 1/4 cup parsley in a blender. Blend until smooth; transfer to a bowl and refrigerate until chilled, at least 30 minutes.
In a medium saucepan, combine the shrimp, bay leaves and peppercorns; add enough water to cover. Bring the water to a simmer and cook the shrimp just until pink, about 4 minutes. Fill a large bowl with ice water and set a medium bowl in the ice bath. Drain the shrimp and transfer the hot shrimp to the inner bowl. Cool the shrimp completely, and then peel and de-vein (discard the shells and tails).
In small glass jars, divide the lettuce and lime segments. Toss the mayonnaise mixture with the chilled shrimp in a bowl. Divide the shrimp among the jars. Top the shrimp mixture with the anchovy croutons. Serve chilled.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.