Place the chicken, mushrooms, celery, carrot, onion, lemongrass, Thai basil, ginger, Thai chile, lime peel, bay leaves, and peppercorns in a large pot. Cover with cold water, approximately 2 to 2 1/2 quarts. Bring the mixture to a boil, and then reduce the heat to a simmer, uncovered. Skim the top, removing the impurities throughout the cooking process. Cook for 1 hour at a low simmer. Remove the chicken from the pot. Remove the leg meat, pull into pieces and reserve for soup garnishes. Remove the remaining chicken and reserve for personal use. Strain the stock into a clean pot.
For the Thai basil chicken soup garnishes: Bring the stock to a boil. Add the chile, if using, sugar, fish sauce, and lime juice. Season the mixture with salt. Add the pulled chicken meat, tomato, and cucumber. Serve the soup, piping hot, garnish with green onion, sesame oil, lime, and Thai basil.
For the egg crepes: Place 2 tablespoons of melted butter in a bowl. Add the ginger, eggs, milk, sesame oil, and salt, and whisk well. Heat a nonstick saute pan over medium heat. Dab a paper towel in the remaining melted butter and coat the pan. Ladle a thin layer of crepe batter in the pan, coating the bottom evenly. Once the batter begins to bubble in the middle, flip the crepe and cook for 3 to 5 seconds on the other side. Remove the crepe from the pan and lay on a dry kitchen towel. Repeat the process with the remaining crepe batter.
For the crepe filling and assembly: Heat the oil in a saute pan over medium heat. Saute the shiitake mushroom caps over the medium heat until tender, about 5 minutes. Season the mushrooms with salt, and pepper, set aside.
Place one crepe on the cutting board. Place 2 spears of asparagus lengthwise, on one end of the crepe. Place a single layer of shiitake mushrooms over the asparagus. Roll the crepe beginning at one end. Place another crepe on the cutting board and roll the finished crepe inside. Cut the crepe on a diagonal into 3 pieces.
Place the Thai basil in a blender and pour the melted butter over. Pulse quickly until just incorporated, ensuring not to bruise the Thai basil. Transfer the mixture to a bowl, place in the freezer for 5 minutes. Remove the butter from the freezer and whisk the mixture until well incorporated, return the Thai basil butter to the freezer. Once the butter has solidified, approximately 5 minutes, whisk again.
Transfer the butter to a container and place in the refrigerator or freezer. Spread the Thai basil butter over the toasted baguette pieces.