Heat olive oil in a skillet over medium heat. Add chopped onion and cook until softened, about 3-4 minutes.
Add minced garlic to the skillet and cook for an additional 1 minute until fragrant. Remove from heat.
In a blender, combine tomatillos, cooked onion and garlic, diced green chilies, chicken broth, ground cumin, and dried oregano. Blend until smooth.
Place chicken breasts in the slow cooker. Pour the blended tomatillo sauce over the chicken.
Season with salt and black pepper.
Cover and cook on low for 6 hours or until the chicken is tender and easily shredded.
Remove chicken breasts from the slow cooker and shred them using two forks.
Return shredded chicken to the slow cooker and stir in chopped cilantro and lime juice.
Mix well and cook uncovered on low for an additional 10 minutes to combine flavors.
Serve hot with your choice of rice, tortillas, or toppings.