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Rinse and dry the chicken pieces and gizzards (reserve livers for another use). Place the chicken and all of the remaining poaching ingredients in an 8-quart Dutch oven or heavy-bottomed pot and cover the solids with cold water. Cover pot and bring mixture to a boil over medium-high heat. Reduce heat to very low and simmer chickens until tender but not dry, about 30 minutes. Use a slotted utensil to remove the chicken to a large bowl to allow the pieces to become cool enough to handle. Remove the meat from the chicken, tear into chunks and set aside for the soup. Return the chicken carcasses, including skin and all other parts, to the pot of poaching ingredients and use to replenish your supply of stock for use at a later date (see the note following this recipe).
Melt the butter in a 10-inch deep-sided skillet over medium heat. While it melts, tear off a sheet of waxed paper (large enough to cover the bottom of the pot) and brush some butter on one side of the paper. Stir the leeks, carrots, celery and parsnips or turnips into the melted butter. Toss to combine to lightly coat the vegetables with butter, and then place the greased side of the waxed paper directly on top. Reduce heat to very low and let vegetables sweat until wilted and only slightly tender, 8 to 10 minutes.
In an 8 to 10-quart soup-pot, bring stock to a boil and reduce heat so stock just simmers. Add the sweated vegetables to the stock and simmer (covered) over very low heat until crisp tender, about 15 minutes. Remove from heat.
Stir the chicken chunks into the soup pot along with the chopped tomatoes, peas, shredded spinach, if using, and chopped parsley. Add salt and freshly ground pepper to taste.
Reheat the soup gently, but thoroughly, until very hot. Taste for seasoning and adjust before serving. Ladle piping hot soup and vegetables into warmed bowls. Serve immediately.
Return the poaching mixture with the chicken bones to a boil. Reduce heat to low and simmer uncovered until the stock is rich with a golden color, 1 to 2 hours. As stock simmers, occasionally press hard on the solids to extract all of their goodness.
Although the addition of butter will produce a richer tasting soup, you can omit it entirely by adding the fresh vegetables directly to the stock instead of sweating them.
The stock can (and should) be made way ahead and, after removing the fat, be stored in the freezer.
Any leftover soup can be frozen in securely covered, heavy-duty freezer containers for several months. (Be sure to label the containers with both the date and contents.) Thaw overnight in the refrigerator and reheat very gently.
The vegetables, as well as the poached chicken for the soup can be prepared (assembled) a day ahead and kept in separate, well-covered bowls, in the refrigerator.
The soup can be fully assembled 1 day ahead, cooled, covered and refrigerated. To refrigerate the soup in the pot, pull a clean kitchen towel over the top and then apply the lid. The towel will prevent any accumulated condensation from falling into the soup and diluting the flavor.