Halve the yellow onions through the stem ends and remove the peels. Cut lengthwise into 1/4-inch thick slices. You should have about 10 cups of onions total.
In a large Dutch oven or similar large, deep pot, melt the butter over medium-high heat. Once the butter melts, add the onions.
Cook, stirring the onions every minute or so, until they soften, about 8 minutes.
Stir in 1/2 teaspoon of the salt. Reduce the heat to medium low to caramelize the onions.
Continue cooking, stirring periodically, until the onions are a rich, even golden brown color. Adjust the heat as needed so the onions become meltingly tender and reduce in volume significantly but do not burn (depending upon how your pot conducts heat, you may need to reduce the heat to low for a while); if at any point the juices on the bottom of the pot start to stick, add 1 to 2 tablespoons of water and use a wooden spoon to scrape the bottom of the pot as needed. The length of time onions take to caramelize will vary based upon your pot, stove, and onions (enamel takes longer than cast iron or stainless steel). Plan on about 45 to 60 minutes.
Deglaze the pot: Increase the heat to medium high. Pour in the wine and use a wooden spoon to scrape up any stuck-on bits. Let simmer 1 minute.
Add the stock, bay leaf, pepper, and remaining 1 teaspoon salt. Increase the heat to bring to a boil, then reduce the heat to maintain a gentle but steady simmer. Continue simmering uncovered for 30 minutes more. Stir in the brandy. Taste and adjust seasoning as desired (depending upon your stock, you may want a bit more salt). Fish out and discard the bay leaf.
Meanwhile, make the toasted bread: Place a rack in the upper third of your oven and preheat to 450 degrees F. Line a baking sheet with parchment paper. Brush both sides of the bread slices lightly with olive oil. Arrange on the prepared baking sheet and toast on the upper rack until lightly browned, 5 to 7 minutes. Set aside.
Just before serving, flip each slice of bread over and sprinkle with 1/4 cup of the cheese. Return to the oven and toast until the cheese is melted and golden, about 2 to 3 minutes. Ladle the soup into bowls, top with a bread slice, and serve with a sprinkle of chopped chives.
*If using regular or even reduced sodium beef stock instead of unsalted, reduce the salt added in step 5 to 1/2 teaspoon; salt at the end to taste.
If you prefer not to cook with alcohol, simply leave it out! You can use part of the beef stock to deglaze the onions.
TO COOK IN INDIVIDUAL CASSEROLE DISHES: Instead of toasting the bread separately, ladle the cooked soup into individual broiler-safe dishes. Lay a bread slice on top (no need to brush with oil) and sprinkle with cheese. Broil 1 to 2 minutes, until the cheese is melted and browned.
TO STORE: Refrigerate soup in an airtight storage container for up to 3 days. Store bread separately in an airtight storage container at room temperature.
TO REHEAT: Rewarm leftovers in a pot on the stovetop over medium-low heat or in the microwave. If desired, toast leftover bread on a baking sheet in the oven at 350 degrees.
TO FREEZE: Freeze soup in an airtight, freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.