In a bowl, combine the turkey, chives, dill, parsley and mustard; season with salt and pepper. Form 4 patties.
In a nonstick skillet, heat the EVOO, 2 turns of the pan, over medium-high heat. Add the burgers and cook through, turning once, 12 to 14 minutes.
Meanwhile, in a saucepan, melt the butter over medium heat. Add the shallots, carrots and celery and cook until softened, 7 to 8 minutes; season with salt and pepper. Sprinkle with the flour and stir for 1 minute, then whisk in the stock and cook until thickened, 3 to 4 minutes. Stir in the peas and heat through.
Top each toasted muffin bottom with a burger and some potpie gravy. Cover with the muffin tops and serve.