Mix all ingredients, marinate and refrigerate quail for at least 6 hours; overnight is ideal. Season the quail and grill until done; this will take about 20 minutes. Be careful not to burn the skin. Place on top of the braised papaya. Garnish with chili oil. (Recipes for the braised papaya and chili oil to follow).
In a saucepan, melt butter and sugar together. Add papaya and cook for 5 minutes. Season. Add lime juice and chicken stock and simmer for 30 minutes or until papaya is al dente. Season with salt and pepper.
Heat chili and cumin in a saute pan until barely smoking. Whisk in oil and salt. Transfer to tall, glass jar and let stand overnight. Oil will separate.