For the remoulade, combine all the ingredients in a bowl, stirring until smooth and well blended. Let it sit for at least 30 minutes to allow the flavors to develop—though it can be used immediately, the sauce deepens in flavor over time.
In a large skillet, pour enough peanut oil to create a 1/4-inch layer and heat over medium-high. To test if the oil is ready, drop in a small pinch of flour—if it sizzles instantly, it's hot enough for frying.
In a wide bowl, whisk together the flour, cornmeal, Cajun seasoning, and salt. Working in small batches, coat the shrimp first in the beaten eggs, ensuring each piece is fully covered, then dredge them in the seasoned cornmeal mixture. Shake off any excess coating before frying.
Carefully place the shrimp in the hot oil and fry until golden brown on both sides, approximately 2 minutes in total. Once crisp and cooked through, transfer the shrimp to a plate lined with paper towels to drain any excess oil.
To assemble the sandwiches, slice each French roll open without cutting all the way through. Generously spread the remoulade sauce on both the top and bottom halves of the bread.
Layer shredded lettuce on the bottom half, followed by the crispy shrimp. Arrange 3 to 4 slices of tomato on top, then gently press the top of the sandwich down to slightly compact it.
Serve immediately with additional hot sauce on the side and, for those who appreciate a classic pairing, a cold beer.