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