Cook the rice first and set it aside.
In a small bowl, stir together the soy sauce, brown sugar, garlic, ginger, and sesame oil.
Heat a large skillet over medium heat. Add the ground beef and cook until it is no longer pink, breaking it up as it browns.
Pour the soy sauce mixture over the beef. Stir and let it simmer for 3 to 4 minutes until the sauce thickens slightly and coats the meat. Remove from heat and stir in the green onions.
While the beef cooks, make the quick pickles. Place the cucumber and carrots in a bowl. Add the rice vinegar, sugar, and salt. Toss to coat and let them sit for at least 10 minutes, stirring once or twice.
To serve, divide the rice among four bowls. Spoon the beef over the rice and top with the drained pickled vegetables. Sprinkle with sesame seeds.