Heat 1 tablespoon oil in now-empty Dutch oven over medium heat. Add onion, carrots, celery, salt and pepper.
Cover and cook, stirring occasionally, until just tender (about 5 minutes). While vegetables are cooking, shred chicken into small bite-size pieces.
Transfer cooked vegetables to bowl with chicken; set aside.
Heat butter in empty Dutch oven over medium heat. When foaming subsides, stir in flour and cook 1 minute.
Slowly whisk in reserved chicken broth and milk. Bring to simmer and stir until sauce thickens, about 1 minute. Add salt and pepper to taste.
Remove from heat and stir chicken-vegetable mixture and peas into sauce.
Pour mixture into 9x13 pan (or if you have small individual-sized oven-safe dishes, you can use those-- see picture above).
Scatter crumble topping evenly over filling.
Bake at 400-degrees until filling is bubbling and topping is well browned (about 12-15 minutes). Serve hot and enjoy!