Place the chicken breasts in the bottom of the slow cooker in a single layer.
Drizzle the olive oil over the chicken, then sprinkle the minced garlic, dried rosemary, salt, and black pepper evenly across the top.
Pour the chicken broth around the edges of the chicken so it pools at the bottom.
Add the trimmed green beans on top of the chicken.
Cover and cook on low for 6 hours or on high for 3 to 4 hours until the chicken reaches 165°F and the green beans are tender.
Remove the chicken and green beans from the slow cooker and serve together with some of the cooking juices.