Heat olive oil in a skillet over medium heat. Add diced onion and cook until softened, about 3-4 minutes.
Add minced garlic to the skillet and cook for 1 minute until fragrant. Remove from heat.
Place the chicken breasts in the bottom of the slow cooker.
Pour the cooked onion and garlic over the chicken.
Add salsa verde, white beans, diced tomatoes with green chilies (including liquid), chicken broth, ground cumin, chili powder, dried oregano, black pepper, and salt to the slow cooker.
Stir gently to combine all ingredients around the chicken breasts.
Cover and cook on low for 6 hours, or until the chicken is cooked through and tender.
Remove the chicken breasts from the slow cooker and shred them with two forks.
Return the shredded chicken to the slow cooker and stir to combine.
Add chopped fresh cilantro and lime juice to the chili, stirring well.
Taste and adjust seasoning with additional salt or lime juice if desired.
Serve the salsa verde chicken chili hot, garnished with extra cilantro if preferred.