Position a rack in the center of the oven and heat the oven to 400°F. Put the peppers on a rimmed baking sheet and roast in the oven, turning every 15 minutes, until browned and wrinkled all over, 45 to 60 minutes. Remove the peppers from the oven, cover with a dishtowel, and set aside to cool. Seed, peel, and cut the peppers into quarters.
Heat the oil in a 4-quart saucepan over medium heat. Add the onion, jalapeño, and rosemary and cook, stirring occasionally, until the onion starts to brown, 8 to 10 minutes. Stir in the peppers and any accumulated juices, broth, sugar, 1-1/2 tsp. salt, 1/4 tsp. black pepper, and 1-1/2 cups water. Bring to a simmer over medium-high heat, cover, reduce the heat to low, and simmer for 5 minutes to blend the flavors. Remove from the heat and let cool slightly.
Purée the soup in batches in a blender or food processor. Strain the puree through a fine sieve into a bowl, using a ladle to push as many of the solids through as possible. Discard the solids in the sieve and refrigerate the soup for at least 3 hours or overnight.
Once chilled, season the soup to taste with salt and pepper. Serve in chilled bowls, sprinkled with the chives. Finish each serving with a drizzle of olive oil.