Heat the oil in a large, heavy saucepan. Add the onion and cook for 5 minutes until softened, add the chicken and cook until lightly browned. Add the potatoes, broth, and plenty of freshly ground black pepper, then bring to a boil. Cover and simmer for 30 minutes until the potatoes are tender and the chicken is cooked. Can be frozen at this point.
Add the broccoli, mesclun, peas, and scallions, stir well, then bring to a boil again. Cover and cook for 5 more minutes, stir in the pesto and reheat.