Preheat an oven to 450 degrees F (230 degrees C).

Mix together the feta cheese, green onion, and tarragon in a small bowl; season with salt and pepper and set aside.

Form the ground turkey into 8 even-sized patties. Top four of the patties with even portions of the cheese mixture. Top each with one of the remaining patties. Pinch the edges together to seal.

Bake the patties in the preheated oven until cooked through, 20 to 30 minutes, turning once.

Heat the olive oil in a skillet over low heat. Cook and stir the red onion in the hot oil until browned; remove the onion and discard. Stir the flour into the oil and cook until deep brown in color. Remove the pan from the heat and add the chicken broth, red wine, parsley, minced onion, bay leaf, and thyme. Return the pan to heat and bring to a boil while stirring; season with salt and pepper. Pour the sauce over the patties to serve.