Make the quick pickled veggies: In a large bowl, toss the matchstick carrots and thinly sliced cucumber with 2 tablespoons rice vinegar, sugar (if using), and 1/2 teaspoon salt; set aside and toss occasionally while you cook.
Prepare green onions: Thinly slice the bunch and separate into about 1/2 cup white/light green bases and 1/2 cup darker green tops.
Heat 1 tablespoon toasted sesame oil in a large nonstick pan over medium-high heat.
Add the ground beef or turkey and cook, breaking it up, until it starts to brown, about 3 minutes.
Stir in the white/light green portion of the sliced green onions and continue to cook until the meat is fully browned; drain excess liquid if needed.
Add minced garlic and ginger paste and cook, stirring, for about 30 seconds until fragrant.
Stir in 3 tablespoons soy sauce, 2 tablespoons honey, 2 tablespoons rice vinegar, and 1/4–1/2 teaspoon red pepper flakes; simmer until the sauce is mostly absorbed and the meat is caramelized, about 2–3 minutes.
Remove the pan from heat and stir in the remaining 1 tablespoon toasted sesame oil, the green onion tops, and sesame seeds if using.
To serve: Divide cooked rice among bowls, top with the seasoned ground beef and a generous pile of pickled veggies, and offer gochujang, chili-garlic sauce, or sriracha on the side.
