Taste test the stems occasionally to check if they are cooked through. Then turn the heat burner back up to a medium heat setting, and add in the tablespoon of sesame oil.
Wait unto right before you cook to chop up the beet leaves. This is because the beet leaves seem to act like spinach and spoil quickly when damaged or cut. During cooking, the stems tend to soak up a lot of water so keep checking and adding more water to cook the stems down as needed. The stems also take much longer to cook compared to the leaves, so it is beneficial to chop the stems more finely to reduce the cooking time.