Mix beef, egg, breadcrumbs, paprika, oregano, thyme, cayenne and 1/2 teaspoon salt. Shape into sixteen even meatballs.
Heat oil in a deep 12-inch skillet over medium-high heat. Brown meatballs for 5 to 6 minutes, turning twice; transfer to a plate, keeping their raw centers for the covered cooking step.
Reduce heat to medium and cook onion, celery and pepper in the skillet for 5 minutes. Add garlic and rice and stir for 1 minute.
Add tomatoes with their juices, broth, okra and the remaining 1/4 teaspoon salt. Bring to a steady simmer, then arrange the meatballs over the rice.
Cover tightly, reduce heat to low and cook for 20 to 23 minutes without stirring, until rice is tender and meatballs reach 160°F at their centers. Add 2 tablespoons water and cook covered for another 3 minutes if the rice is still firm and the pan is dry.
Remove from heat and rest covered for 5 minutes. Fluff the rice around the meatballs and serve four meatballs with each portion.