Preheat the broiler to high.
Heat a deep pot over medium to medium-high heat. Add EVOO, once around the pan, and butter to the pot. Add the onions to the pot as you slice them and then the chopped garlic. Season with salt and pepper. Add rosemary sprigs and a bay leaf.
Cook the onions for 20-25 minutes, stirring frequently, until tender, sweet and deep caramel-colored. If the onions are burning in spots before browning all over, add a splash of water and stir every now and then, scraping the bottom of the pot.
While the onions are cooking, make the grape tomato topping: In a small bowl, combine the grape tomatoes, basil, a little salt and pepper.
Under the broiler or toaster oven, toast the bread and rub both sides of the slices with the crushed garlic clove.
Make sure the toast circumference is smaller then the bowl in which you are serving it in. Place a piece of cheese on top of each toast and pop them under the hot broiler until the cheese melts and bubbles.
Once the onions are tender and brown, add the balsamic vinegar and white wine, stirring up all the brown bits from the bottom of the pot. Add 6 cups of stock and cover the pot to bring the soup up to a quick boil.
Arrange 4 small, deep soup bowls or crocks on a cookie sheet. Once the soup reaches a boil, remove the bay leaf and rosemary stalk, and ladle into bowls. Float the cheesy toast on the soup and top each toast with a mound of the grape tomato mixture.