In a small saucepan, combine the carrots, water, and salt. Bring to a boil, reduce the heat to medium, cover, and cook until the carrots are tender, about 10 minutes. Uncover and increase the heat to high to evaporate the water completely. When no water remains, add the milk. Purée the carrots and milk in a blender until smooth.
In a clean pan, melt the butter. Add the flour, stirring until smooth. Cook over medium heat, stirring constantly, for 2 to 3 minutes. Add the puréed carrots, bring to a boil, and cook, stirring, until thick, 3 to 4 minutes. Pass the mixture through a mesh strainer into a 2-qt. bowl. Add the chives, ginger, lemon juice, and honey. Taste and adjust the seasonings.
Whisk the egg yolks one at a time into the carrot mixture. In a large, clean bowl, whisk the egg whites to firm but not dry peaks. With a rubber spatula, fold half of the egg whites into the carrot mixture, mixing very gently. Fold in the other half until just combined. Don’t overmix or you’ll deflate the egg whites.
Pour the soufflé mixture into the gratin dish. Run your thumb around the inside rim to help a cap form. Place the dish gently into the oven and bake until browned and puffy, about 15 minutes. It should be set everywhere but still somewhat creamy in the center. Serve immediately.