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1
Peel the carrots, wash them in cold water, and slice them into 3/8 inch disks. The thin tapered ends can be cut thicker. Choose a saute pan that can contain the carrot rounds spread in a single snug layer, without overlapping. Put in the carrots and butter, and enough water to come 1/4 inch up the sides. If you do not have a single pan large enough, use two smaller ones, dividing the carrots and butter equally between them. Turn on the heat to medium. Do not cover the pan.
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2
Cook until the water has evaporated, then add salt and the 1/4 teaspoon of sugar. Continue cooking, adding from 2 to 3 tablespoons water as needed. Your objective is to end up with well-browned, wrinkled carrot disks, concentrated in flavor and texture. It will take between 1 and 11/2 hours, during which time you much watch them, even while you do other things in the kitchen. Stop adding water when they begin to reach the wrinkled, browned stage, because there must be no liquid left at the end. In 30 minutes or a little more, the carrots will become so reduced in bulk that, if you have been using two pans, you will be able to combine them in a single pan.
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3
When done - they should be very tender - add the grated Parmesan, turn the carrots over completely once or twice, transfer them to a warm platter, and serve at once.
