Combine kasha and egg in a small bowl; stir well.
Heat oil and butter in a large saucepan over medium heat. Add onion; cook 5 minutes or until soft. Add kasha mixture; cook 1 minute. Stir in salt and broth; bring to a boil. Cover, reduce heat, and simmer 15 minutes or until kasha is tender.
Remove from heat; fluff with a fork. Stir in dill.