In a small skillet, heat 1 tablespoon EVOO, 1 turn of the pan, over medium heat. Add the carrot, celery, onion and garlic and cook until tender, about 5 minutes. Transfer to a bowl and let cool. Reserve the skillet. Mix the turkey, dill and poultry seasoning into the vegetable mixture; season with salt and pepper. Form into 4 patties.
In the reserved skillet, heat the remaining 1 tablespoon EVOO over medium heat. Add the patties and cook, turning once, until firm, about 12 minutes; transfer to a plate. Add the butter to the skillet to melt, then stir in the hot sauce. Add the patties to the sauce, turning to coat. Top each with some of the blue cheese and a cheddar slice. Tent with foil, turn off the heat and let stand until the cheese is melted.
Top each muffin bottom with some lettuce, a patty, some relish and a muffin top.