1. Coarsely shred celery root in a food processor fitted with the shredding attachment. Transfer to a medium bowl. Add mayonnaise, crème fraîche, vinegar, mustard, shallot, and 2 tbsp. chives and mix well. Season to taste with 1/2 tsp. salt and 1/4 tsp. pepper. Set aside while you cook the chicken.
2. Slice each chicken breast in half horizontally to make 4 thin chicken scaloppine. Season chicken with remaining 1/2 tsp. each salt and pepper.
3. Set a large (not nonstick) frying pan over medium-high heat; when hot, add oil. Cook chicken until browned, 2 1/2 to 3 minutes. Turn and cook until fully cooked through, 2 to 3 minutes.
4. Top toasts with chicken and spoon on celery root salad. Slice each toast in half and sprinkle salad with more chives if you like.
*To prep celery root, cut off the ends, then set the root flat. Following the curve, cut off all the peel. If you're not going to use it soon, set the trimmed root in a bowl of water with a couple of tablespoons of lemon juice added to prevent it from turning brown.