Preheat the broiler. Rub the onion halves all over with 2 teaspoons vegetable oil and place cut-side up on a small foil-lined baking sheet. Broil until completely charred on top, about 12 minutes; set aside. Preheat the oven to 275 degrees F.
Meanwhile, heat the remaining 1 tablespoon vegetable oil in a large Dutch oven over medium-high heat. Add the pork and cook, turning, until browned on all sides, about 8 minutes. Add 3/4 cup water, the soy sauce, mirin, 2 scallions, one-quarter of the ginger and 3 smashed garlic cloves. Bring to a boil, cover and transfer to the oven. Cook, turning the pork every hour, until tender, 2 1/2 to 3 hours. Transfer the pork to a plate or cutting board. Skim off the fat from the cooking liquid, then strain through a fine-mesh sieve into a bowl; set aside.
Meanwhile, make the stock: Bring 10 cups cold water, the chicken wings and bacon to a boil in a large pot. Boil vigorously for 3 minutes, then skim off any foam and reduce the heat to a gentle simmer; cook 1 hour. Add the broiled onion, dried shiitakes, remaining ginger, 4 scallions and 4 smashed garlic cloves. Simmer 1 more hour, skimming occasionally as needed. Strain the stock through a fine-mesh sieve into another pot; you should have about 8 cups (add more water if necessary). Stir in the strained pork cooking liquid; season with salt. Keep hot over low to medium heat.
Bring a medium saucepan of water to a boil. Gently lower the eggs into the water and adjust the heat to maintain a gentle simmer. Cook 7 minutes for soft-boiled or 10 minutes for medium-boiled, then drain and run the eggs under cold water to stop the cooking. Peel the eggs and set aside.
Bring a large pot of water to a boil. Add the noodles and cook as the label directs. Drain well.
Thickly slice the pork. Carefully halve the eggs lengthwise. Divide the 3 minced garlic cloves and the noodles among bowls. Ladle the hot stock into each bowl. Top with the sliced pork, an egg half and assorted toppings.