Tie a bouquet garni of rosemary, thyme, and bay leaves with kitchen twine, leaving about a 10-inch 'tail' on one end of the string. Set aside. Thinly slice two onions. Also set aside.
Melt 3 tablespoons of unsalted butter in a 4 or 5-quart Dutch oven over medium-high heat. Reduce the heat when the butter becomes foamy. Lightly brown the butter, add the onion slices, then stir well to coat. Stir occasionally, letting them achieve a golden hue over 6-8 minutes.
The onions are ready for the soup when softened and lightly caramelized. Using the long 'tail,' tie the aromatic bundle of herbs to the Dutch oven handle so the herbs will be easy to remove later. Use a wooden spoon to submerge the herbs in the onions. Pour the entire bottle of hard cider over the onions and herbs. Bring to a boil, reduce heat, and simmer for 8 to 10 minutes.
Finely grate some hard cheese. Use what resonates with your eaters. I am using a mix of Beemster, Gruyere, and Parmesan. Set aside. Measure out a handful of prepared croutons. Also set aside.
The remaining 2 tablespoons of butter await its turn in a separate medium saucepan. Melt it over medium heat, then whisk in the ¼ cup all-purpose flour. This flour-butter mixture or roux should be stirred continuously to achieve a deep caramel color. As the roux cooks, it cooks out the smell and taste of raw flour. The final appearance should be deeply caramel in color and the aroma nutty and pleasant. Slowly whisk in the 2 ¾ cups chicken stock. Let this mixture boil, reduce the heat, and allow it to simmer until slightly reduced, around 15 minutes.
Remove the bouquet garni, reserving a single bay leaf. Pour the onion and cider blend into the stock. Add the reserved bay leaf and stir well. If using a rind of hard cheese to impart flavor, add it now. Simmer over low heat for 15 minutes.
Line a sheet pan with parchment paper. Set each soup crock on the sheet pan. Adjust your oven rack so it is about 6 inches from the broiler element. Set the broiler to high. Ladle soup into the crocks almost to the top, but not quite. Top each with croutons and cheese. Drizzle a tablespoon or two of heavy cream over the cheese.
Place the prepared cookie sheet with your soups on the top rack under the broiler. MONITOR! Broil them until the tops are golden, and the soup beneath is bubbly, about 2 minutes.
Remove the sheet pan with the soups from the oven. Allow the soups to stand for 5 minutes before using oven mitts to place each bowl on a plate lined with a napkin or paper towel to avoid the bowl sliding. THE BOWLS WILL BE HOT. Garnish with fresh herbs if desired and serve.