For the pickled shrimp: Combine the lemon juice, vinegar, garlic, onions, pickling spice, chiles and 1/2 cup water in a medium saucepan. Bring to a boil and cook for 1 minute. Reduce to a simmer; cook for another 4 minutes. Transfer to a large bowl and let cool to room temperature.
Fill a bowl with ice water; set aside. Bring a medium pot of salted water to a boil. Add the shrimp and cook for 1 1/2 minutes. Drain and plunge in the ice water to stop cooking. Drain again and pat dry. Add the cooked shrimp to the pickling liquid, cover and refrigerate for at least 4 hours and up to 24 hours.
Drain the shrimp well and put in a bowl; add the hot sauce and toss to combine.
For the remoulade deviled eggs: Remove the yolks from the eggs and place in a medium bowl; mash to a paste with a fork. Place the egg whites, cut-side up, on a large plate or platter.
Add the mayonnaise to the mashed yolks; mix until well combined. Fold in the mustard, cornichons, dill, chives, 1/4 teaspoon salt and 1/8 teaspoon pepper.
Fill the egg whites with the yolk mixture. Top each deviled egg with a pickled shrimp and garnish with a parsley leaf. Sprinkle the shrimp with a pinch of smoked paprika.