In a large nonstick skillet over medium-high heat, quickly brown the meat, breaking it apart with a wooden spoon, but don’t cook it all the way through. Season with salt and pepper, to taste. Drain off any rendered grease and transfer the ground meat to the slow cooker. Mix in the onion, carrot, and bell pepper.
In a shallow bowl, whisk together the garlic, ginger, juice, soy sauce, gochujang, honey, and rice vinegar. Pour over the meat in the slow cooker and stir to combine. Cover and cook on low for four hours.
While you wait for the sloppy joe filling to cook, prepare the quick pickled toppings. Combine the red onion, radish, and cucumber in a wide, non-reactive bowl.
In a small saucepan over medium heat, combine the rice vinegar, water, sugar and salt. Stir and heat until the mixture is hot and the sugar is fully dissolved. Pour the hot mixture over the sliced onion, radish, and cucumber, and stir in the red pepper flakes. Cover and refrigerate until ready to serve.
After four hours, combine two tablespoons of cornstarch and two tablespoons of water in a shallow bowl. Mix until well combined and pasty. Pour into the slow cooker and mix, then increase heat to high. Cook, covered, until the sauce has thickened, about 30 more minutes.
Serve the Korean Sloppy Joes in buns or bibb lettuce leaves with a sprinkle of toasted sesame seeds and the pickled vegetables, to top.