Preheat oven to 180 degrees C. Line a baking tray with foil.
Peel away any excess paper from the bulb of garlic then cut the top off to expose the cloves. Rub exposed cloves with 1/4 teaspoon olive oil and set onto the prepared baking tray. Poke the eggplant all over with a fork; place onto baking tray.
Bake in preheated oven until the garlic has turned golden brown and the eggplant is tender; 30 to 40 minutes. Allow to cool until cool enough to handle. Peel eggplant and chop into large chunks. Peel or squeeze the roasted garlic from its skin and set aside.
Heat 1 tablespoon olive oil in a large saucepan over medium heat. Stir in onion and cook until soft and translucent; about 5 minutes. Pour in chicken stock then stir in reserved eggplant, roasted garlic, tomato puree and cayenne pepper. Bring to a boil over medium-high heat then reduce heat to medium-low. Cover and simmer for 40 minutes.
Once cooked carefully puree soup in batches in a blender or food processor until smooth. Return soup to the stove in a clean saucepan. Stir in milk, season to taste with salt and pepper; cook over medium-low heat until hot.