Heat 1 tablespoon oil in a large pot over medium-high heat. Add ground chicken and cook until browned, breaking up meat into small chunks with a wooden spoon. Add garlic, ginger, white ends of scallions, Thai chile, soy sauce, oyster sauce, and sugar. Stir to combine and simmer for 1 minute.
Add chicken broth and sliced mushrooms. Cover pot and bring to a boil. Reduce heat to low and simmer for 10 minutes.
Add Thai chili jam, miso, and mirin, stirring to dissolve miso paste. Add halved bok choy and simmer for another 5 minutes or until bok choy is tender.
Meanwhile, cook noodles according to directions on package. Drain and portion noodles into 2 bowls. Ladle soup into bowls and top with Thai basil, chopped green scallions, and fried eggs. Serve immediately.