Caramelize the onions: Peel, halve, and thinly slice 3 large yellow onions. Heat 3 tablespoons extra virgin olive oil in a large 12-inch (30 cm) skillet over medium heat. Add the onions, ¾ teaspoon salt, ¼ teaspoon black pepper, and 2 teaspoons sugar, and cook for 5 minutes, stirring often. Pour in ½ cup water, reduce the heat to medium-low, and cook for about 20 minutes, stirring occasionally, until the water has evaporated and the onions are soft and golden. Turn the heat to high, stir in 2 tablespoons balsamic vinegar, and cook for 2 minutes.
Add the beans: Stir in 3 cloves garlic (grated) and cook for 30 seconds. Add 2 cups vegetable broth, 2 cans butter beans, and 1 teaspoon dried thyme. Bring to a gentle simmer and cook uncovered for 10 minutes. For a creamier broth, mash a few of the beans with the back of a spoon.
Finish: Turn off the heat and let the skillet sit for 1 minute, until it stops bubbling. Stir in ½ cup grated parmesan until the broth looks creamy. Taste and add more salt and pepper if needed.
Serve: Serve hot with toasted baguette, sourdough, or other crusty bread for dipping into the rich onion broth.
