In a deep skillet, heat the EVOO, 2 turns of the pan, over medium-high heat. Add the carrots, onion and celery, season with salt and pepper and cook until softened, 8 to 10 minutes. Add the peas and chicken and cook until the chicken is cooked through, about 4 minutes; set aside.
Preheat the broiler. In a small, heavy saucepan, bring 1 cup chicken broth and 1/2 cup cream to a boil. Add the polenta and cook, whisking, for 3 minutes. Lower the heat, stir in the gruyere until melted and set aside.
Push the chicken and vegetables off to the side of the skillet. Over medium heat, add the butter to melt, then stir in the flour; cook for 1 minute. Gradually whisk in the remaining 2 cups chicken broth and 1/2 cup cream, stir in the chicken and vegetables and bring to a simmer. Stir in the tarragon and mustard; season with salt and pepper.
Place four ovenproof soup bowls on a baking sheet and pour in the chicken mixture. Scoop some polenta on top to cover completely. Broil until the tops are browned, 2 to 3 minutes.