Preheat the oven to 350°F (175°C). Line a large sheet pan with parchment paper and set aside.
Dice the cooked chicken into bite-sized pieces. Finely dice the shallot and celery. Shred the carrots using the large holes of a grater. Shred the cheddar if not pre-shredded.
In a large bowl, combine the diced chicken, shallot, celery, shredded carrots, and 1 cup of the shredded cheddar.
In a separate medium bowl, whisk together the sour cream, mayonnaise, Cajun seasoning, garlic powder, paprika, and black pepper until smooth.
Pour the dressing over the chicken mixture and gently stir until evenly combined. Taste and adjust seasoning, adding more Cajun seasoning if desired.
Slice the French bread into thick slices about 1 to 1½ inches thick and arrange them on the prepared sheet pan.
Spoon about 1/3 cup of the chicken mixture onto each bread slice and spread into an even layer with the back of a fork.
Top each melt with about 1½ tablespoons of the remaining shredded cheddar so the leftover 1 cup is used for toppings.
Bake in the preheated oven for 8–10 minutes, or until the cheese is melted and slightly bubbly. Remove from the oven.
Garnish as desired with fresh herbs, thinly sliced Roma tomatoes, or a drizzle of hot sauce, then serve immediately.