Heat the olive oil in a large skillet over medium-high heat. Add the pork shoulder pieces and brown them on all sides for about 6 minutes total. Transfer the pork to the slow cooker.
Add the diced onion and minced garlic to the same skillet. Cook for 3 minutes until the onion softens, then scrape everything into the slow cooker.
Pour in the chicken broth. Add the drained white beans, salt, black pepper, and rosemary sprigs. Stir once to combine.
Cover and cook on low for 7 hours. The pork should be fork-tender by this point.
Remove the rosemary sprigs. Stir in the chopped kale. Cover again and cook on low for 20 more minutes until the kale wilts.
Taste and adjust salt if needed. Serve hot.