Wash, peel, and core quinces, reserving cores and peels. Coarsely chop flesh and transfer fruit to a large pan. Wrap cores and peels in cheesecloth, tie the bag with kitchen string, and add it to the pan.

Pour in enough water to cover quinces and boil, with the pan partly covered, until fruit is soft, about 30 to 40 minutes. Remove the bag of peels and pass quince flesh through a sieve or food mill. You should have about 2 ½ pounds of fruit pulp.

Transfer quince pulp to a saucepan and add sugar. Cook and stir over low heat until sugar is dissolved. Continue cooking, frequently stirring with a wooden spoon, until paste becomes very thick and has a deep orange color, about 1 ½ hours.

Lightly grease a 9x13-inch baking dish or line it with greased parchment paper. Transfer quince paste to the baking dish, spreading it about 1 ½ inches thick. Smooth the top and allow it to cool.