In a small skillet, heat 1 tablespoon EVOO, 1 turn of the pan, over medium heat. Add 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 back to the skillet, turning to coat in the sauce. Top each with some of the blue cheese and a slice of cheddar. Tent with foil, turn off the heat and let stand until the cheese is melted.