Cook the chicken, onion, chili, garlic, salt, pepper, allspice, and oil in a heavy pot over low heat until tender, about 2 hours.
Remove chicken to a bowl, discard bones, and shred. Add cooked onions and garlic to the chicken and drizzle with honey.
Blend jalapeños, cilantro, celery leaves, garlic, lime juice, olive oil, and salt until smooth.
Blend avocado, celery greens, mayonnaise, lemon juice, salt, pepper, and olive oil until smooth. Stir in chives.
Assemble tacos with chicken, schug, and avocado sauce. Serve warm.