Brush any dirt off of mushrooms and remove stems. Heat oil in a saucepot over medium-low heat. Add garlic and thyme and allow the garlic to permeate the oil until fragrant, about 5 minutes. Remove the garlic and thyme and add the mushrooms. Gently stir the mushrooms in the oil and cook for 5 minutes. Remove from heat, stir in vinegar and salt and allow the mushrooms to cool completely in the oil. Use for serving immediately or store in the refrigerator in a sealed container for up to a week.
Prepare a hot direct fire for grilling.
Place the oxtail and chicken wings on a rimmed sheet pan. Drizzle 2 Tbsp of oil over the meat and toss to coat. Place under the broiler until the oxtail and chicken wings are thoroughly browned. Flip and continue to broil until browned on the other side. Reserve until needed.
In a large bowl, toss to coat the leek, scallions, and onion with 2 Tbsp of oil. Place on the grill and grill until charred on all sides. Reserve.
In a large heavy-bottomed pot, heat the remaining 1 Tbsp of oil over medium-high heat, sauté the ginger cut side down until caramelized. Add spices and thyme and stir until fragrant, about 2 minutes. Next, add oxtail, chicken, leek, scallions, onion, chicken stock or water, fish sauce, sugar, lime zest, and juice. Bring to a boil over high heat then reduce to a low simmer. Simmer for 4 hours or until stock has reduced by a third. Strain out the solids with a fine meshed sieve or cheesecloth and chill quickly in a large bowl placed in an ice bath. Remove any fat that rises to the surface. Reheat broth to serve or refrigerate in a sealed container for up to three days or freeze for up to a month. Alternatively, a pressure cooker can be used for this step by sautéing the ginger and spices over medium heat, adding the remaining ingredients, and cooking for 1 hour with full (high) pressure. Allow to depressurize naturally for a clear broth.
To serve, place a serving of mushroom conserve in each bowl and pour hot pho stock over. Garnish with fresh herbs and shaved black truffle (if desired).