Whirl 1 cup pineapple with 3 cups of water until smooth, about 1 minute. Strain liquid into a 4 qt. pot.
Add broth, ginger, garlic, and verbena leaves. Simmer over medium heat, covered, until flavors have melded, 10 to 15 minutes.
Strain mixture again, discard solids, and return to pot. Add remaining 2 cups pineapple along with the remaining ingredients and cook over medium heat until chicken is firm, about 10 minutes.