Preheat oven to 300 degrees F.

Cut six 1 to 1 1/2-inch-thick pieces of baguette long enough to span the width of six broilerproof soup crocks. Set on a baking sheet and lightly toast in oven, checking often, until dried out, about 45 minutes. Remove from the oven and set aside.

Melt butter in a 6-quart wide-bottom Dutch oven or other heavy wide-bottom enameled pot set over medium-high heat. (The wider the surface area, the more the onions come in direct contact with the heat, and therefore the better; this encourages them to caramelize.)

Add onions, salt, and pepper and continue to cook, stirring as needed with a wooden spoon or flexible heat-resistant rubber spatula, 30 minutes. Add thyme bundle and continue to cook for 10 minutes, scraping bottom regularly. (After onions have cooked for 40 minutes, you should begin to see them caramelize. At this point it will happen faster and faster, so you will need to stir onions frequently to prevent burning for these 10 minutes, incorporating all browned bits until the onions are golden brown.)

Reduce the heat to low and add brandy. Continue to cook and stir until brandy has evaporated. Add beef and chicken broth, bring to a boil, then reduce to a simmer and continue to cook, 25 minutes. Turn off heat and discard thyme bundle.

Set oven rack as close to broiler element as your soup crocks will allow and turn on broiler. Set soup crocks on a rimmed baking sheet. Ladle in soup. Top with a crouton and a generous amount of cheese. Broil soup for 2 to 3 minutes, checking often, until cheese is golden brown and bubbly. Serve immediately, being sure to warn guests that bowls are very hot.