Pour half of the oil into a large skillet over medium heat. When the oil is hot, add the leeks, salt, and pepper and cook, stirring occasionally, for 5 minutes, or until softened. Add the wine or water, broth, and herbs. Bring to a boil and let bubble for 1 or 2 minutes.
Add the chicken, reduce the heat to medium-low, cover, and simmer for 5 or 6 minutes, or until the meat is barely cooked through. Remove the chicken from the skillet.
Add the potatoes and bring to a boil. Reduce the heat so the liquid bubbles enthusiastically. Cook for 5 minutes, or until the potatoes are almost tender.
Stir in the carrots or parsnips and cook for another couple of minutes. By now, the liquid should be thickening. If not, raise the heat and cook another couple of minutes, stirring to prevent the vegetables from sticking. Add the remaining oil gradually, stirring vigorously with the back of a spoon as you do so.
Add the peas and asparagus to the skillet. Cook, stirring occasionally, for 3 minutes, or until the vegetables are brightly colored and just tender. Chop or slice the chicken and return it to the skillet, along with any juices that have accumulated. Add the lemon juice. Warm through, then taste and adjust the seasoning. Serve in shallow bowls, garnished with parsley.