Choose a sauté pan that can hold all of the carrots in one layer, without overlapping. Add the carrots, butter, and enough water to come up ¼" on the sides.
Turn the heat to medium and leave the pan uncovered.
Cook until the water has evaporated, then add salt and sugar.
Continue cooking, adding 2-3 tbsp of water as needed.
After 1- 1½ hours, the carrots should look well-browned and wrinkled. When they reach this stage, do not add any additional water. They should be very tender and will have greatly reduced in size.
To prepare this recipe in advance, stop here before adding the Parmesan. Then, when reheating, add it in right before serving.
Remove from the heat and toss with the grated Parmesan. Serve immediately.