Cut each chicken breast half into two portions. Cut through at the meatiest portion of each breast so you end up with four relatively equal-sized portions.
In a deep-sided, heavy-bottomed pot, brown the chicken pieces in the vegetable oil over medium-high heat, turning once to brown on both sides.
Reduce the heat slightly, remove the chicken pieces to a plate, pour in the broth, stirring and scraping to dissolve all of the caramelized chicken juices.
Return the browned chicken to the pot, and add the green onions, lemon slices, ginger root, chile-garlic sauce, and black pepper. Cover, and cook for 30 minutes.
Add the chopped sweet onion, cover, and continue simmering just until the chicken is tender and cooked, about another 10 or 15 minutes.
Adjust the seasonings, adding salt if desired.
The chicken may be prepared to this point up to 24 hours ahead and refrigerated.
Five or ten minutes before serving, bring the mixture to a boil, then remove the chicken pieces with a slotted spoon. Add the angel hair pasta, cook until done (this only takes about 3 minutes). When the noodles are cooked, return the chicken pieces to the pot and heat through.
Adjust seasonings. If you remember, fish out any remaining pieces of lemon (it will be almost disintegrated by now, though) before serving. Serve in large soup bowls or pasta bowls.