Preheat an oven to 350 degrees F (175 degrees C). Place the bacon in a large, deep skillet, and cook over medium-high heat, turning occasionally, until evenly browned but still soft. Reserve the bacon grease in the skillet, and cool the bacon slices on a paper towel-lined plate. Once cool, mix the bacon together with the feta cheese, 3 tablespoons of sour cream, oregano, and black pepper in a small bowl; set aside.
Lay a chicken breast flat onto your work surface. Use the tip of a sharp boning or paring knife to cut a 2-inch pocket in the chicken breast. Repeat with the remaining chicken breasts. Spoon the bacon mixture into the pockets. Pour the flour, egg, and bread crumbs into separate, shallow dishes. Gently press the chicken breasts into the flour to coat. Dip each into the beaten egg, then press into bread crumbs.
Reheat the bacon grease over medium heat. Brown the chicken breasts on both sides in the hot fat, about 2 minutes per side. Reserve the bacon grease in the pan. Place the breasts on a baking dish, and bake in the preheated oven until the chicken is no longer pink and the filling is hot, 20 to 25 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Meanwhile, place the potatoes into a large pot and cover with salted water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain.
While the potatoes are boiling, cook the onion in the remaining bacon grease over medium heat until very tender and golden brown, about 10 minutes. Once the potatoes are done, mash together with the onion, butter, and remaining 3 tablespoons of sour cream. Serve the chicken breasts accompanied by the mashed potatoes.