Boil rice and water in a saucepan. Cover; cook on low 18 minutes. Rest covered off heat 10 minutes.
Cover eggs with cold water by 1 inch in a separate saucepan. Boil, cover, remove from heat, and rest 11 minutes. Chill in ice water 5 minutes; peel and halve.
Heat oil in a large skillet over medium. Cook onion, carrot, and celery 7 to 8 minutes until softened.
Increase to medium-high; add beef and cook 7 to 8 minutes, breaking into fine crumbles, until 160°F. Stir in garlic, ginger, and curry powder 1 minute.
Add apple, tomatoes, ketchup, Worcestershire, soy sauce, and broth. Simmer uncovered over medium-low 8 to 10 minutes, stirring, until a spatula briefly leaves a path.
Divide rice among four bowls; spoon curry on one side. Add two egg halves to each; serve hot.