Place chicken thighs in a large resealable bag; pour in blue cheese dressing. Coat chicken with dressing, squeeze out excess air, and seal the bag. Marinate in the refrigerator for 4 hours to overnight.
Preheat oven to 420 degrees F (216 degrees C). Spray a 9×13-inch baking dish with cooking spray.
Heat 2 tablespoons olive oil in a skillet over medium heat; cook and stir celery in the hot oil for 1 minute. Add onion; cook and stir until onion is glossy and softened, about 5 minutes. Stir in garlic; cook for 30 seconds. Transfer onion mixture to the prepared baking dish; return skillet to burner.
Pour remaining 1 tablespoon olive oil into the skillet; cook and stir rice in the hot oil until fragrant, 2 to 3 minutes.
Mix chicken broth, rice, and vinegar into the onion mixture in the baking dish. Remove chicken from marinade and arrange in the baking dish, creating 2 rows of chicken thighs side by side. Discard remaining marinade. Sprinkle each chicken thigh with Cajun seasoning. Coat chicken thighs with 1 cup panko crumbs. Cover baking dish with aluminum foil.
Bake in the preheated oven for 35 minutes. Remove aluminum foil from dish. Mince the reserved celery leaves. Sprinkle chicken thighs with remaining 1/2 cup panko crumbs and Parmesan-Romano cheese; sprinkle minced celery leaves over chicken.
Continue to bake chicken until no longer pink at the bone and the juices run clear, about 10 more minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).
Turn on the oven’s broiler and broil chicken until browned and crisp, 1 to 3 minutes. Remove from oven and let cool for 10 minutes before serving.