In a large saucepan over medium-high, melt 2 tablespoons butter. Add the onion, carrots and ½ teaspoon salt; cook, stirring occasionally, until the vegetables are lightly browned, 4 to 6 minutes. Add 1 teaspoon cumin seeds, ¼ teaspoon cayenne and ¼ teaspoon black pepper; cook, stirring, until fragrant, about 2 minutes. Add the chickpeas and the 1 cup reserved liquid, the garlic, 4 cups water and ¼ teaspoon salt. Bring to a boil over medium-high, then reduce to medium-low, cover and cook, stirring occasionally, until the garlic is soft when the head is squeezed with tongs, 30 to 35 minutes.

Remove the pot from the heat. Squeeze the garlic cloves from the head into the chickpea mixture; discard the empty skins. Let the chickpea mixture cool for about 5 minutes.

Meanwhile, in a small saucepan over medium-high, melt the remaining 3 tablespoons butter. Add the remaining 2 teaspoons cumin seeds and cook, stirring, until fragrant and the butter has stopped foaming, 1 to 1½ minutes. Add the sesame seeds and another ¼ teaspoon cayenne (if using); cook, stirring, until the sesame seeds are toasted and fragrant, about 1½ minutes. Remove from the heat and set aside, covered.

Using a blender and working in batches so the jar is never more than half full, puree the chickpea mixture until smooth. Return the soup to the pan and cook over medium-low, stirring often, until warmed through, 2 to 5 minutes. (Alternatively, use an immersion blender to puree the soup directly in the saucepan.) Taste and season with salt and black pepper. Serve drizzled with the butter mixture.