For the Lamb Croquettes:

1. In a large pot, combine the lamb neck, beef or lamb broth, onion, carrots, and minced garlic. Bring to a boil over high heat, then reduce the heat to low and let it simmer for about 2 hours until the lamb neck is tender and easily falls apart. 
2. Once the lamb neck is cooked, remove it from the pot and let it cool slightly.
3. Shred the lamb neck into small pieces using a fork or your fingers, discarding any bones or tough connective tissues. Reserve the braising liquid for later. 
4. In a separate saucepan, melt the butter over medium heat. Stir in the flour and cook for 2-3 minutes until bubbly and fragrant. Gradually whisk in the milk, stirring constantly to avoid lumps. Cook for another 2-3 minutes until the sauce thickens.
5. Add the shredded lamb neck, salt, black pepper, ground nutmeg, and chopped parsley to the sauce. Stir well to combine and let it cool for a few minutes.
6. Once the lamb mixture is cooled, shape it into small patties or balls, about the size of a golf ball.
7. Dip each lamb patty or ball into the beaten eggs, then coat with breadcrumbs, pressing gently to adhere the breadcrumbs to the lamb mixture.
8. Heat about 1 inch of vegetable oil in a frying pan or deep fryer over medium heat.
9. Fry the lamb korokke in batches for about 3-4 minutes per side, or until golden brown and crispy. Remove the lamb korokke from the oil and place them on paper towels to drain excess oil.
10. Serve the lamb korokke hot, with the Japanese BBQ sauce 

For the Japanese BBQ Sauce:

1. In a small saucepan, combine the soy sauce, water, brown sugar, rice vinegar, mirin, sesame oil, grated ginger, minced garlic, and chopped green onions. Stir well to combine. 
2. Place the saucepan over medium heat and bring the mixture to a boil, then reduce the heat to low and let it simmer for about 5 minutes, stirring occasionally. 
3. In a small bowl, whisk together the cornstarch and water to make a slurry. Slowly pour the cornstarch slurry into the saucepan while stirring constantly to thicken the sauce. 
4. Continue to simmer the sauce for another 2-3 minutes until it reaches your desired consistency. If the sauce is too thick, you can add a little water to thin it out. 
5. Remove the saucepan from the heat and let the Japanese BBQ sauce cool to room temperature.
6. Once the sauce is cooled, transfer it to a jar or airtight container and store it in the refrigerator for at least 30 minutes to let the flavors meld together. 
7. Use the Japanese BBQ sauce as a marinade for your favorite meats, such as beef, pork, chicken, or vegetables. Simply coat the meat or vegetables in the sauce and let it marinate for at least 30 minutes, or overnight for more intense flavor. 
8. Grill, broil, or pan-fry the marinated meat or vegetables until cooked through, basting with extra sauce while cooking for additional flavor. 
9. Serve on the Japanese croquettes with a glass of Hoc.