Bring rice and 1 3/4 cups water to a boil in a saucepan. Cover, reduce heat to low and cook for 18 minutes, then rest covered off heat for 10 minutes. Spread no more than 1 inch deep on a clean tray and refrigerate promptly for at least 1 hour until cold.
Heat 2 tablespoons oil in a 12-inch nonstick skillet over medium heat. Press cold rice into an even layer about 1/2 inch thick and cook undisturbed for 9 to 11 minutes until golden underneath.
Turn rice in broad sections and cook for 5 to 7 minutes more until crisp in patches. Transfer to a plate and keep warm.
Heat the remaining tablespoon oil in a separate large skillet over medium-high heat. Add broccoli and cook for 4 minutes to brown several surfaces, then add 1/4 cup water, cover for 2 minutes and uncover to evaporate any liquid; transfer to a bowl.
Brown beef in the broccoli skillet for 7 to 8 minutes, breaking it into fine crumbles and cooking to 160°F. Stir in garlic for 30 seconds.
Whisk miso, vinegar, honey and 3 tablespoons warm water until smooth. Stir into beef with butter and cook for 1 minute, then fold in broccoli. Divide crisp rice into four bowls and spoon the beef mixture over part of each portion.