In a large, heavy-bottomed pot, sauté onions in oil over medium-low heat until very brown and soft, about 40 minutes. Don’t rush; this is the key to this soup: onions that are tender and sweet, not burned or bitter. Stir onions every few minutes. Season with salt and pepper.

When onions are done, add caraway seeds and stir for a few seconds. Add Port and turn up the heat to a light simmer. With a wooden spoon, scrape off all the caramelized onion bits stuck to the bottom of the pot. Continue to cook, stirring often, until most of the liquid is evaporated and you are left with a sauce-like ragout.

Add miso to a bowl and stir in 1/2 cup stock. Stir until the paste is completely incorporated into the liquid. Add to onions and simmer lightly until half the liquid is evaporated. Add remaining stock and simmer lightly for about 15 minutes.

Preheat oven to 400°F. Place four oven-proof bowls on a baking sheet. Pour about 1 cup soup into bowls, leaving room on top for cheese and bread. Add 2 tablespoons cheese, then a bread slice (if the slice is too big, cut it into smaller pieces and wedge in the bowl). Top with 2 more tablespoons cheese, and bake until cheese is browned and bubbly. Serve immediately.