Heat the olive oil in a large skillet over medium-high heat. Add the onion, carrots, and celery. Saute until tender. Add the flour and ¼ cup of the chicken broth.
Stir until the mixture thickens and then slowly add the rest of the chicken broth. Cook over medium heat until the sauce thickens, about 3 minutes. Add the chicken, peas, salt, and pepper.
Preheat the oven to 400 degrees F. Pour the chicken mixture into a baking dish.
To prepare the dumplings—in a medium bowl add flour, baking powder, and salt. Whisk until well combined.
Add egg and milk and mix with a wooden spoon until just combined. Drop the dough over the chicken mixture with a spoon (the dough will make about 8 dumplings). Place into the oven and bake for 15 minutes or until the dumplings are golden brown.