In a large mixing bowl, combine the shredded cabbage, white onion, and minced green onions.
In a separate small bowl, whisk together the distilled white vinegar, lime juice, kosher salt, pepper, and white sugar until the salt and sugar are fully dissolved.
Pour the vinegar mixture over the shredded cabbage and onions in the large mixing bowl.
Toss the ingredients together thoroughly, ensuring that the cabbage is well coated with the dressing.
Taste the slaw and adjust the seasoning if needed. You can add more vinegar, salt, or sugar according to your taste preference.
Pack all of the cabbage mixture into a 2 quart jar with a tight fitting lid and place in the refrigerator for 15 minutes to several hours, or best overnight, before using. This resting time allows the flavors to meld together and for the cabbage to slightly soften.
Before serving, shake, taste and adjust seasonings according to your preference.