1. Combine chicken stock, chicken breast, onion, bay leaves, potatoes, and guascas in a large saucepot or Dutch oven. Add more water if necessary to cover solids by 1-inch. Bring to a boil over high heat. Reduce to a simmer and cook until chicken breast is cooked through, about 10 minutes. Transfer chicken breast to plate and set aside until cooled to room temperature.
2. Continue to simmer potatoes until russets and papas Criollas are completely tender and falling apart, about 45 minutes longer, adding more water as necessary. Discard onion and bay leaves. Using whisk, mash potato pieces against side of pot and stir vigorously to thicken soup. The soup should be thick, with large chunks of potatoes still remaining.
3. Add corn cobs and continue to cook, stirring occasionally, until corn is cooked and soup is thickened to the consistency of thick heavy cream, about 15 minutes longer. Remove from heat.
4. While soup is cooking, make the aji: Combine onion, cilantro, and chile pepper in small serving bowl and add water until just covered. Season generously with salt and stir to combine. Set aside. Discard chicken skin and bones. Shred meat into bite-size pieces and set aside in serving bowl.
5. Season soup to taste with salt and pepper. Serve immediately, passing picked chicken, chopped capers, sour cream, and aji table-side for diners to add to their bowl. Serve avocado slices on the side.