Preheat the oven to 400 degrees F (200 degrees C).
Heat 2 tablespoons butter and olive oil in a skillet over medium heat. Add carrots, celery, and onion. Add 1/2 teaspoon garlic powder; season with salt and pepper. Cook and stir until starting to soften, about 3 minutes. Add chicken; cook until heated through, about 3 minutes more. Add frozen peas; cook until turning bright green, about 1 minute. Stir and remove from heat.
Melt 1/4 cup butter in a saucepan over medium-high heat until bubbling, about 2 minutes. Reduce heat to medium and add 1/4 cup plus 3 tablespoons flour. Whisk quickly to remove all lumps, about 1 minute. Cook, whisking slowly, until gravy is thickened, about 2 minutes more.
Whisk chicken broth into the gravy thoroughly. Bring to a boil, stirring occasionally until thickened, about 5 minutes. Add the remaining garlic powder and onion powder. Season with salt and pepper. Add the vegetable-chicken mixture; stir and pour into a 9x13-inch glass baking dish.
Combine 3 cups flour, 1 cup milk, and 1/2 cup plus 1 tablespoon mayonnaise in a bowl. Add remaining milk gradually until mixture is sticky; you may have leftover milk. Drop spoonfuls of dumpling batter to cover the top of the filled baking dish.
Bake in the preheated oven until dumplings are light golden brown, about 25 minutes.