Preheat the oven to 400 degrees F. Have ready four large ramekins or a 9 × 13-inch baking pan.
Heat the oil in a large, heavy-bottomed saucepan over medium-high heat. Add the chicken and sauté until almost cooked through, 5–7 minutes. Add the potatoes, carrots, celery, green beans, sweet peas, and garlic, and mix the vegetables in with the chicken. Season with the Cajun seasoning and sauté until the vegetables begin to soften and the chicken is fully cooked, about 5 minutes. Using a slotted spoon, transfer the chicken and vegetable mixture to a large bowl, keeping the oil and drippings in the pan.
Turn the heat to medium and add the flour to the pan, whisking it into the oil. Cook, stirring constantly, until a blonde roux forms, 5–7 minutes. You want to make sure the raw flour taste cooks out. Add the chicken broth and whisk to get rid of any lumps. Whisk in the heavy cream. Add the pink peppercorns, scallions, parsley, and thyme and
season with salt and pepper as desired. Bring the sauce to a light simmer over medium- low heat and cook until the sauce has thickened. You will know when desired thickness is reached when there is minimal dripping from the back of your wooden spoon.
