Into a large saucepan over medium heat, add the remaining 1 tablespoon olive oil. When the oil is hot, add the onions then season with salt and pepper, to taste. Cook for 3 minutes, stirring frequently, until onions are softened.
Add the eggplant and cook for 1 minute.
Add the minced garlic and cook for an additional 1 minute.
Add the vegan chicken-style broth and bring to a boil. Reduce the heat to medium-low and simmer for 10 minutes.
Using a handheld immersion blender, or working in batches with a countertop blender, puree the soup until its consistency is smooth.
Stir in the oat milk and butter. When the butter has melted, let the soup simmer for 3 minutes. Adjust seasoning to taste with salt and pepper.