Heat oil in a 6-qt saucepan over medium-high heat. Brown wings (8 minutes). Add scallions, garlic, ginger and water and bring to a boil.
Reduce heat to medium-low, and simmer until chicken falls off the bone, (1 hour). Pour broth through a fine strainer, discard solids and set broth aside.
Meanwhile, place mushrooms in a large bowl and pour 4 cups boiling water over. Let sit until softened (15 minutes). Drain mushrooms, reserving soaking liquid. Finely chop mushrooms and set aside.
Heat oil in a 10 inch skillet over medium-high heat. Working in batches, add potatoes and fry until lightly browned and cooked through (10 minutes). Transfer to paper towels to drain and set aside.
Add mirlitons to oil and fry until soft (30 seconds). Transfer to paper towels to drain and set aside.
Drain off all but 2 tbsp of oil and return to medium heat. Add eggs and swirl skillet to make an omelette. Cook, turning once, until just cooked through (2 minutes). Transfer to a cutting board. Roll up omelette like a cigar and cut crosswise into 1/4 inch thick strips. Set aside.
Return saucepan to medium-high heat. Add bacon and cook, stirring, until crisp (14-16 minutes). Add scallions, garlic, and ginger and cook, stirring, until soft (8 minutes). Add mushrooms and cook until liquid is evaporated (6 minutes).
Add reserved broth and mushroom soaking liquid. Bring to a boil. Add noodles and cook, stirring occasionally until al dente (7 minutes). Stir in soy sauce and vinegar.
To serve, divide noodle soup among 8 large serving bowls and top each with potatoes, mirliton, and egg strips. Serve with chile paste.