Cut the leeks in half lengthwise, then cut them crosswise into half-moons. Fill a large bowl with cold water. Add the leeks to the bowl and separate all the layers to release their grit. Pull the leeks from the water and dry on paper towels.
Season the cutlets with salt and pepper and dredge in flour. Heat a large nonstick skillet over medium-high heat, add 2 tablespoons of the EVOO, 2 turns of the pan, and cook 4 of the cutlets until cooked through, 3 to 4 minutes on each side. Transfer the cooked cutlets to a plate and cover loosely with foil. Add the remaining EVOO, 1 turn of the pan, to the skillet and repeat with the remaining 4 cutlets. When the cutlets are cooked, add 2 tablespoons of the butter to the skillet and when it has melted, whisk in the 2 tablespoons of flour and cook for 1 to 2 minutes. Whisk in 2 cups of the chicken broth and season the gravy with salt and pepper. Turn the heat to low.
While the first batch of cutlets is working, heat a deep skillet over medium heat and add 2 tablespoons of the butter. Add the leeks, celery and apple, season with salt and pepper and cook, stirring occasionally, for about 5 minutes. Crumble the pumpkin muffins into the vegetable-apple mixture, season with the poultry seasoning and mix together the stuffing. Fold in the pistachios, then moisten the stuffing with some of the remaining broth until moist and crumbly.
In another deep skillet, add about 2 inches of water and some salt and bring to a simmer. Drop in the green beans and cook until just fork-tender, about 4 to 5 minutes. Drain and toss with the remaining 1 tablespoon of butter.
To serve, mound the pumpkin-muffin stuffing on plates and then top each mound with 2 turkey cutlets. Ladle the gravy over the turkey and serve the buttered green beans alongside.