Peel, halve, and thinly slice 3 pounds yellow onions (about 10 cups).

Melt 4 tablespoons of the unsalted butter in a Dutch oven or large heavy-bottomed pot over medium heat. Add the onions, 1 teaspoon dried thyme, 2 dried bay leaves, 1 ½ teaspoons of the kosher salt, and ½ teaspoon black pepper. Cook, stirring often and scraping the bottom of the pot with a wooden spoon to prevent burning, until the onions are soft and amber brown, 35 to 45 minutes.

Meanwhile, prepare the following, adding each to the same medium bowl as you complete it: Peel and cut 3 medium carrots crosswise into ¼-inch-thick rounds (about 1 ¼ cups). Cut 3 medium celery stalks crosswise into ¼-inch-thick pieces (about 1 cup). Mince 3 garlic cloves.

Add the remaining 2 tablespoons unsalted butter to the onion mixture. When the butter is melted, add the carrot mixture and cook, stirring occasionally, until the carrots are crisp-tender and brightened in color, about 5 minutes. Meanwhile, season 2 pounds boneless, skinless chicken breasts or thighs all over with the remaining 1 teaspoon of the kosher salt. (If using thick chicken breasts, start at the thickest end and hold your knife parallel to the cutting board, then cut the chicken breasts in half — also known as butterflying.)

Pour ¾ cup dry white wine and 1 tablespoon dry sherry into the Dutch oven. Cook, stirring occasionally, until mostly evaporated, 2 to 3 minutes. Sprinkle with ¼ cup all-purpose flour and cook, stirring constantly, for 1 minute to cook the floury taste out.

While stirring constantly, pour in 6 cups low-sodium chicken broth. Add the chicken in a single layer. Increase the heat to medium-high and bring to a lively simmer.

Reduce the heat as needed to maintain a gentle simmer. Cook, stirring and scraping the bottom of the pot occasionally, until the chicken is cooked through and an instant-read thermometer inserted into the thickest part of the chicken registers at least 165°F, about 10 minutes for thighs or 12 to 20 minutes for breasts. Meanwhile, prep the cheesy toasts if desired.

Arrange an oven rack 3 to 4 inches below the broiler unit and set the oven to broil. Cut ½ baguette crosswise on a bias into 1-inch-thick slices (about 8). Place in a single layer on a rimmed baking sheet. Grate 2 ounces Gruyère cheese on the largest holes of a box grater (about heaping ½ cup). Sprinkle the bread evenly with the cheese.

Using tongs, transfer the chicken to a clean cutting board. Use two forks to shred the chicken into bite-sized pieces. Return the chicken to the pot. Remove and discard the bay leaves. Taste and season the soup with more kosher salt as needed.

Broil the bread until the cheese is melted and bubbling, 30 seconds to 1 minute. Top each bowl of soup with a piece of cheese toast.