Use a slicer or chef's knife to trim and thinly slice the onions. The slices should be about 3/16-1/4 inch / 4-6mm wide from root to stem.
Heat the oil in a large, heavy-bottomed pot over medium high heat. Add the onions and salt. At first, the onions will crowd the pot and fill it almost completely. Stir the onions in the pot to help distribute the oil, then cover them and cook, covered, for 5 minutes. Uncover and stir. Reduce the heat to medium and re-cover the pot.
Continue cooking the onions in the covered pot until they're caramelized. This will take 40-60 minutes in total. Uncover the onions and stir them every 10 minutes throughout, adjusting the heat if necessary as you go. Each time you stir the onions, you'll notice that onions toward the bottom of the pot are browning a bit. Stir well to disperse those caramelized bits—they're good! However, if the onions start burning considerably, lower the heat. In the end, you want the onions to be a deep golden brown throughout, tender and reduced considerably in size.
Add the garlic to the pot. Cook, stirring as you go, for 1-2 minutes, or until the garlic is fragrant. Add the white wine to the pot and allow it to come to a simmer. Continue cooking for 5 minutes, stirring often, or until the liquid has mostly evaporated.
Sprinkle the flour lightly over the onions and stir. Continue cooking and stirring for another couple minutes, or until the flour is evenly absorbed and the onions have thickened. Slowly pour in the broth, 1 cup / 240ml at a time, stirring as you go. When all the broth is added to the pot, add the bay leaves and vinegar and cover the pot. Turn the heat to high and bring the soup to a simmer, then turn the heat back to low. Simmer the soup, covered, for 30 minutes.
Remove the bay leaves from the pot. Taste the soup; adjust the salt as needed and add freshly ground black pepper to taste.
Turn your broiler to high. Place your bread slices on a foil-lined baking sheet and toast in the oven for 2 minutes per side, till they're just barely golden. Remove the bread from the oven and spread each slice with a generous layer of the homemade cashew cheese. Alternatively, arrange your store-bought vegan cheese over the bread slices. Return the bread to the oven and broil for another 3-4 minutes, or until the cheese is set and turning golden (homemade cheese) or melted (store-bought cheese).
Divide the soup into bowls and top each bowl with a slice of the cheese toast. It's fine if the slice of toast sinks into the soup—in fact, that's the most delicious way to enjoy the bread here!
The soup can be stored in an airtight container in the fridge for up to 1 week and frozen for up to 6 weeks.
Notes
*You can use red, white, yellow, or sweet onions to make the soup.