I did a huge amount of research on Babylonian food (plus I had a friend who specialised in their culture). The Mesopotamians very kindly wrote down recipes for all sorts of dishes. We know what they ate and how they ate it. Sausages were very common - mostly goat, but there was beef too, and I conjecture that there was probably chicken sausages as well.
When the billy got a bit old, other males left alive were chosen to be the new billy for a while. Most male goats are a surplus to requirement. A good goatherd recognised which males are possible breeding stock, and which are for eating. If there were a lot of female goats, some of those became dinner too.
In general, there's just the one dominant "billy" goat. That's how goats operate. Sure, there's sparring with other males trying to be alpha, but the alpha billy is the one tending to the nanny goats.
Obviously they had no ketchup, but they certainly had mustard plants, and they sowed and reaped garlic and onions (to make fried garlic/onions), as well as wheat and rye (for the bun).
One of their recipes was garlic and onions sauteed and then pureed with a bit of salt, and maybe some fish sauce (i.e. pureed anchovies mixed with olive oil). Dried fish were commonly traded all over the place, including pilchards and anchovies. With the anchovies they may not have needed any salt. These would make an interesting addition to a sausage inna bun.

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Some more notes about Babylonian cooking, courtesy of my friend (who, alas, I've lost touch with)...
Ancient Babylonian cuisine was rich and varied, including meat from cows, sheep, goats, pig, deer and fowl, as well as eggs, fish, shellfish and even turtles. Stews were common, with Akkadian records providing 21 different meat stew recipes and four different vegetable stew recipes.
Among the vegetables eaten by the ancient Babylonians were beets, peas, arugula, lettuce, turnips, legumes (such as chickpeas) and mushrooms. They were especially fond of using onions in their cooking, along with all of its related plants, such as garlic, scallions and leeks. Grains were also common, including pearl barley and oats.
Fruits enjoyed by the ancient Babylonians included apples, figs and pomegranates.
The ancient Babylonians were particularly interested in flavoring their food and used a range of herbs and spices. These included mint, coriander and chives, among others. Condiments, such as a pickled fish sauce (
siqqu), were also used, while honey, dates, grape juice and raisins were favored for sweetening dishes.
For cooking, a variety of oils and fats were used, such as clarified butter and animal fat, as well as sesame, linseed and olive oils.
The presentation of food was important to the ancient Babylonians, who used a selection of fresh greens, chicken gizzards, pastry crusts and even bird feathers to garnish their meals.
