Marinate the pork patties. In a large mixing bowl, combine the ground pork, minced garlic, minced shallots, fish sauce, soy sauce, brown sugar, honey, chicken bouillon powder, black pepper, and neutral oil. Mix until thoroughly combined. Marinate for at least 1 hour or up to overnight.
Marinate the pork belly. In a separate mixing bowl, combine the sliced pork belly, minced garlic, minced shallot, fish sauce, soy sauce, brown sugar, honey, chicken bouillon powder, and black pepper. Mix until evenly coated. Cover and marinate for at least 1 hour or up to overnight.
Make the quick pickled vegetables. In a mixing bowl, whisk together warm water, vinegar, sugar, and salt until everything dissolves. Then add the green papaya and carrot, making sure everything is submerged. Pickle in the refrigerator for at least 30 minutes or up to 24 hours.
Prepare Dipping Sauce. In a mixing bowl, combine warm water, sugar, fish sauce, and lime juice. Mix until the sugar has completely dissolved. Then add the Thai chilis.
Shape the patties. Divide the ground pork into about 2 1/2-inch-wide x 1/2-inch-thick patties. If you have a grilling basket, brush the basket with oil and place it at least 1 inch apart.
Grill the patties and pork belly. Preheat a grill over medium-high heat to about 450°F-500°F (232-260°C) and lightly oil the grates. Grill the pork patties about 3-5 minutes per side or until charred and cooked through (160°F or 71°C). At the same time, grill the pork belly for 2-3 minutes per side, only flipping once.
(Option 2) Stove cooking: Heat 1 tablespoon of oil in a cast iron or nonstick skillet over medium-high heat. Work in batches. Add the pork patties in a single layer and cook them 3-5 minutes per side, and the pork belly 2-3 minutes per side, or until charred and cooked through.
Assemble and serve. In a bowl, add the grilled pork patties, grilled pork belly, pickled vegetables, and the dipping sauce. Serve with vermicelli noodles, lettuce, and herbs. Enjoy!