Early in the day, make the Remoulade sauce by placing the first 11 ingredients (mayonnaise down to freshly ground black pepper) in a food processor. Process until well combined. Transfer to a non-reactive airtight container. Chill thoroughly before using. This sauce can be kept in the refrigerator for up to two days.
Place lettuce in a large bowl. Drizzle with half of the remoulade sauce and season with salt and pepper. Toss until well combined.
Place tomatoes in a small bowl and drizzle with olive oil, toss to combine.
Divide the lettuce evenly between 4 plates. Place avocado halves on top of the lettuce and divide the tomatoes evenly between the plates. Toss the shrimp with the remaining remoulade sauce and place on top of the avocado. Garnish with the scallions and serve immediately.