In a medium bowl, whisk together ¾ cup of rice vinegar, ½ cup of water, 1 tablespoon of honey, and ¾ teaspoon of salt. Add the thinly sliced cucumbers and mix well. Set aside for at least 10 minutes.

In a large sauté pan, heat 1 tablespoon of olive oil. Add the shredded carrots and ½ teaspoon of salt, and cook for about 10-15 minutes until softened and slightly browned. Transfer to a bowl and keep warm.

In the same sauté pan, add 2 teaspoons of olive oil, and stir in the grated onion, minced garlic, and grated ginger. Cook until fragrant, about 3 minutes.

Add 2 pounds of ground beef, breaking it apart. Cook, stirring occasionally, until browned, about 5-7 minutes. Drain excess fat if needed.

Return the beef to the pan, drizzle with honey, soy sauce, gochujang, and mirin. Stir well and simmer for about 3 minutes until the sauce thickens slightly.

To serve, lay out butter lettuce leaves on plates. Spoon the beef mixture into each leaf, followed by sautéed carrots and pickled cucumbers. Serve with cooked rice if desired.