1. Rinse and dry chilies; cut in half lengthwise and discard stems and seeds. Peel onions; cut crosswise into 1/2-inch-thick slices. Arrange chilies (skin up) and onions in a single layer in a 12- by 17-inch pan.
2. Broil 4 to 6 inches from heat until vegetables are soft and slightly charred, 20 to 25 minutes, turning onions as needed to brown evenly. When chilies are cool enough to handle, pull off and discard skins.
3. In a blender or food processor, whirl half the chilies and onions with about 3/4 cup chicken broth until smooth, scraping container sides as necessary. Rub mixture through a fine strainer set over a bowl; discard residue. Repeat to purée remaining chilies and onions with another 3/4 cup broth and strain into bowl.
4. Stir in remaining 1 1/2 cups broth, half-and-half, and chicken. Cover and chill up to 1 day.
5. In a 3- to 4-quart pan over medium-high heat, stir bisque often until steaming, about 15 minutes. Place avocado, tomato, and cilantro in separate small bowls.
6. Ladle bisque into wide bowls. Serve with avocado, tomato, cilantro, and salt to add to taste.