Heat olive oil in a pot over medium heat and sauté diced onion and garlic until translucent.
Add chicken breast, oregano, cumin, salt, and pepper, searing the chicken for 2 minutes on each side.
Pour in chicken broth and bring to a gentle boil, then reduce heat and simmer for 15 minutes until chicken is cooked through.
Remove chicken, shred it with two forks, and return it to the pot along with the rinsed hominy.
Simmer for another 5 minutes to allow the flavors to meld and the hominy to heat through.
Break the tostada shells into thick strips.
Ladle the posole into a bowl and top with sliced radishes, fresh cilantro, a squeeze of lime, and the crispy tostada strips.