Heat the oil in a large, heavy-bottomed (preferably cast-iron) skillet over medium heat. Whisk in the flour and cook, stirring constantly, until the mixture darkens to a rich, reddish-brown color, 30 to 35 minutes. (Be careful not to let the flour burn or the gumbo will taste burned.) Once the roux reaches the desired color, add the onions, celery and peppers and cook, stirring, 2 to 3 minutes. Add the garlic, thyme and bay leaves, and cook an additional 1 to 2 minutes. Whisk in the beer, stirring to scrape any browned bits on the bottom of the skillet.
Transfer the roux mixture to a large Dutch oven or stockpot over medium-high heat. Add the Chicken Stock, salt, black pepper and cayenne, and bring the mixture to a boil. Reduce the heat and simmer the gumbo for 20 minutes, stirring occasionally. While the gumbo is simmering, slice the sausage lengthwise and then make crosswise cuts, forming 1/2-inch-thick half-moons. Place the sausage in a large skillet over medium heat and cook, stirring occasionally, until the sausage is rendered and lightly browned, 8 to 10 minutes.
Transfer the sausage to a paper-towel-lined plate and pat dry. Add the cooked sausage, okra and tomatoes to the gumbo and simmer an additional 30 minutes, stirring occasionally. Stir in the chicken and cook an additional 10 minutes. Taste and adjust the seasonings, adding more salt or pepper if needed. Just before serving, remove and discard the bay leaves and stir in the green onions. Ladle the gumbo into warmed soup bowls and top with cooked rice. Serve hot.
For the whole chicken: Rinse the chicken under cold running water and pat dry. Place the chicken in a large stockpot or Dutch oven and add 5 quarts cold water, the carrots, celery, onions, garlic, salt, thyme and bay leaf. Bring the mixture to a boil, reduce the heat to low and simmer for 45 to 50 minutes, skimming off any foam that rises to the top. Remove the whole chicken from the water mixture and set aside to cool slightly. Continue to simmer the broth mixture for an additional 30 minutes, skimming occasionally. Strain the broth through a fine-meshed sieve or strainer and discard the solids. The stock can be used immediately or chilled and kept refrigerated for up to three days or frozen for up to one month. The meat can be pulled off of the whole chicken and reserved for another use.
For the chicken bones: Rinse the bones under cold running water and drain. Place the bones in a large stockpot or Dutch oven and add 5 quarts cold water, the carrots, celery, onions, garlic, salt, thyme and bay leaf. Bring the mixture to a boil, reduce the heat to low and simmer for 1 hour, skimming off any foam that rises to the top. Strain the broth through a fine-meshed sieve or strainer and discard the bones and vegetables. The stock can be used immediately or chilled and kept refrigerated for up to 3 days or frozen for up to 1 month. Makes about 5 quarts.