Combine chicken, water, onion, carrot, celery, 1/2 bunch cilantro, garlic, bay leaf, salt, thyme, and black pepper in a large soup pot and bring to a boil, about 5 minutes. Reduce heat and simmer until chicken is no longer pink inside, about 1 hour.

Remove chicken from broth and let sit until cool enough to handle. Shred chicken in a large bowl and reserve for later.

Transfer cooked vegetables and 1 cup broth to a blender, working in batches if needed. Cover and hold lid down with a potholder; pulse a few times before leaving on to blend. Return to the pot.

Increase heat to medium-high and add Yukon Gold potatoes, red potatoes, and russet potatoes to the pot. Add corn cobs and continue cooking soup until potatoes soften and start to dissolve, about 1 hour.

Season ajiaco with salt and transfer to a large serving bowl. Place shredded chicken, crema, 1 bunch chopped cilantro, Mexican cheese blend, diced avocado, chopped hard-boiled eggs, and capers in individual pretty serving bowls to be served with the soup as garnish.