Place the pork shoulder chunks into the slow cooker. In a small bowl, stir together the teriyaki sauce, honey, minced garlic, and grated ginger until combined. Pour the sauce over the pork and turn the pieces to coat them evenly.
Cover and cook on low for 7 to 8 hours or on high for 4 to 5 hours, until the pork is very tender and pulls apart easily with a fork.
Remove the pork from the slow cooker and shred it using two forks. Return the shredded meat to the sauce in the slow cooker and stir to coat it thoroughly.
Spoon the cooked white rice into four bowls. Top each bowl with a generous portion of the saucy pork and an equal amount of steamed broccoli florets.
Finish each bowl with sliced green onions and a sprinkle of sesame seeds before serving.