Preheat oven to 350 degrees.
Place a skillet over medium/high heat and toast the walnuts for 1-2 minutes. Transfer the walnuts to a small bowl and set aside.
Add the tablespoon of extra virgin olive oil to the skillet and place it over medium/high heat.
Add the onions, apples, celery, garlic, red and green bell peppers, and sauté for 7-8 minutes when the skillet is hot. Remove the mixture from the skillet, transfer it into a bowl, and set aside.
Add 2 1/2 cups of the sourdough cubes and 3 cups of the crumbled cornbread into a large bowl. Fold in the onions, apples, celery, garlic, red and green bell peppers mixture, toasted walnuts, dried cranberries, and chopped flat-leaf parsley, and add the seasonings.
Gently fold all ingredients together, and add the vegetable stock one cup at a time. Gently fold the mixture after you add each cup of vegetable stock. The key to great dressing is that it shouldn't be soggy or too dry. However, adjust to your preference. I like my dressing to be moist in the middle but firm and browned on top; it will lose a bit of its moisture while baking in the oven. After all the ingredients are mixed, taste it and adjust the seasoning to your liking.
Spoon the dressing into your preferred baking dish; this should be a high-walled dish about 2 to 4 inches deep.
Drizzle the remaining extra virgin olive oil over the top of your dressing.
Place the dressing onto the center rack of a preheated 350-degree oven and cook for an hour.
Remove from the oven and serve with cranberry sauce or your favorite vegan gravy.