Preheat oven to 375°. Heat 2 tbsp. oil in a 4-qt. pot over medium heat. Add carrot, onion, and celery and cook, stirring often, until softened, about 5 minutes. Transfer mixture to a bowl and add peas; set aside.
Add remaining 1 tbsp. oil to pot. Add flour and whisk until dry and crumbly, about 30 seconds. Pour in milk 1/4 cup at a time, whisking well after each addition until smooth and creamy (mixture will be paste-like at first). Add reserved vegetables, salt, pepper, and chicken to milk mixture, stirring gently to combine. Spoon mixture into a large gratin dish (about 2 1/2 qts.) or ovenproof frying pan. Sprinkle with tarragon.
Mix bread and butter together in a large bowl, tossing well to coat. Top chicken mixture with bread. Sprinkle with cheese. Bake until browned and starting to bubble around the sides, 20 to 25 minutes.
Note: Nutritional analysis is per 1 1/2-cup serving.