Add chicken, chopped onion, garlic, thyme, oregano, red pepper flakes, and bay leaves to pot. Then add water and lock lid. Set manual timer for 30 minutes at high pressure.
Remove stems from collard green leaves and separate leaves into 2 pieces. Discard stems. Cut leaves into 1-inch pieces so that they are easily spoonable. This is easiest by stacking several leaves at a time.
Once cook time is complete, allow pressure to release naturally for about 30 minutes. If there is any pressure left at that point, go ahead and quick-release remaining pressure.
Remove leg quarters and transfer them to a large bowl. Pour broth through a fine mesh strainer and set aside. Discard cooked aromatics.
Add olive oil, onions, celery, carrots, and 1 teaspoon salt to pot. Sweat vegetables for about 5 minutes, stirring frequently.
Add broth to pot, along with 2 teaspoons salt, tomatoes, rinds (if using), collard greens, brown sugar, and black pepper. Mix well, lock lid, and cook for 10 minutes at high pressure.
Meanwhile… remove all chicken meat from bones in large pieces. Discard all skin and bones.
Once cook time is complete, quick-release pressure. Then add chicken and gently mix well. Serve each bowl topped with a spoonful of basil pesto (with cornbread dumplings) & enjoy!