Heat butter in a small skillet over medium heat.

Add diced carrots to the skillet and sauté for about 2 minutes, or until the carrots begin to soften.

Add peas to the skillet and sauté for another minute.

Remove the carrots and peas from the heat and add them to a large bowl.

Add minced cooked chicken breast, mashed potato, onion powder, and ground nutmeg to the bowl.

Season the mixture with salt and pepper, and mix until all ingredients are well combined.

Add egg and flour to the mixture and mix until combined. If the mixture seems too wet, add another tablespoon of flour.

Cover the mixture and chill in the refrigerator for at least 30 minutes to firm up.

Prepare the coating station by placing flour, beaten eggs, and panko breadcrumbs into three separate shallow bowls.

Divide the chilled mixture into 16 equal portions. Use a small cookie scoop to form each portion into a ball.

Coat each croquette in flour, ensuring it's fully covered.

Dip the floured croquette into the beaten eggs, coating it completely.

Roll the egg-coated croquette in panko breadcrumbs, pressing gently to adhere.

Heat the oven to 200 degrees F to keep the croquettes warm.

In a large skillet, heat about 1/2 inch of canola oil until it is shimmering but not smoking.

Fry the croquettes in batches, turning as necessary, until all sides are golden brown, about 3 minutes per batch.

Transfer the fried croquettes to a wire rack set over a baking sheet.

Place the baking sheet with the croquettes into the preheated oven to keep warm while frying the remaining croquettes.

Sprinkle the croquettes with salt and serve immediately with warm gravy dipping sauce.