Place a small stockpot or soup pot on low-medium heat. Add 2 tablespoons of oil to pot and add sliced/chopped onion, and 1 teaspoon each salt and sugar. Cover roughly 5-10 minutes, stirring occasionally until liquid is released. Uncover pot, add half the water and cook over med-low heat until onions begin to caramelize, stirring the onions repeatedly so that no burning occurs. Add more water as needed, continue stirring for about 10-12 minutes or until onions have caramelized into a nice brown color. When done, remove from heat and set aside. Let onions cool for a few minutes.
Transfer onion to Vitamix or Blendtec. Add Greek yogurt (or tufu and vegan mayo), 1 tsp olive oil, salt, and pepper.
Transfer dip to a bowl and add salt and pepper to taste. Add optional soy sauce/tamari and most of the parsley or other options and mix in, sprinkle the rest as a garnish on top.
Chill in the fridge for at least an hour to meld the flavors. Serve with your choice of veggies or chips! *
* Make ahead of time so flavors in dip have a chance to meld!