Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet 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 sheet. Poke the eggplant all over with a fork; place onto baking sheet.
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 broth, 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 half-and-half, season to taste with salt and pepper; cook over medium-low heat until hot.