Cut the apples into quarters without peeling or coring them. Much of the pectin is in the cores and flavor in the peels. Cut out damaged parts.
Cook the apples: Put the quartered apples into a large pot, add the vinegar and water, cover, bring to a boil, reduce heat to simmer, cook until apples are soft, about 20 minutes. Remove from heat.
Push the apples through a food mill: Ladle apple mixture (cooked apples and liquid) into a chinois sieve (or food mill) and using a pestle force pulp from the chinois into a large bowl below.
Add sugar, spices, lemon zest, and juice: Measure resulting apple pulp. Add 1/2 cup of sugar for each cup of apple pulp. Stir to dissolve sugar. Add cinnamon, ground cloves, allspice, lemon zest, lemon juice, and a pinch of salt. Taste and adjust seasonings if necessary.
Cook the apple mixture and stir often: Cook the apple sugar mixture uncovered in a large, wide, thick-bottomed pot on medium low heat, stirring often to prevent burning. Scrape the bottom of the pot while you stir to make sure a crust is not forming at the bottom.
Cook until thick, smooth, and dark brown in color (about 1 to 2 hours). A small bit spooned onto a chilled (in the freezer) plate will be thick, not runny. You can also cook the puree on low heat, stirring only occasionally, but this will take much longer as stirring encourages evaporation. (Note the wider the pan the better, as there is more surface for evaporation.)