Brush an 8" x 8" pan with a small amount of neutral oil and line with parchment paper. Brush the parchment paper with oil.
Core and roughly slice the fruit. Place slices and water in a heavy-bottomed pot with a lid.
Cover and cook the fruit over medium heat for 20 minutes. Uncover and cook for an additional 10 to 15 minutes or until the fruit is very tender and falling apart. (The quince will take a little longer to get tender than the apples will.) If the mixture dries out before the fruit is tender, add another splash of water and put the cover back on the pot.
Remove the pot from the heat. Pass the mixture through a food mill or press it through a fine mesh strainer until skins are removed.
Return the fruit puree to the pot and add the remaining ingredients. Simmer for an hour or more, stirring frequently to keep the bottom of the pot from burning. Keep an eye out for when the mixture begins to mound up. You want the mixture to be as sturdy and thick as possible while still being spreadable.
Once the puree has thickened, spread it into the prepared pan, doing your best to achieve a smooth surface and uniform thickness. Allow to dry overnight.
The next morning, gently touch the surface of the fruit paste. If it's still wet or tacky and the paste is soft, put the pan into a 150 degree F oven or dehydrator. Check every hour or so and remove the paste when the surface is dry to the touch.
Press gently around the pan, especially on any areas that may be thicker than others. If the underside of the paste is still soft and spreadable, put the pan into the fridge to cool for an hour or two. Once it's cooled, you should be able to gently pull up the square of paste and flip it back into the pan, soft side up. Return the pan to the dehydrator or oven, checking every hour or so. Remove when the surface feels dry and the texture has firmed up.
Cut into shapes using a knife or cookie cutters and serve with cheeses, or cut shapes, toss in granulated sugar, and serve immediately as a sweet. Store in an airtight container in the refrigerator.
