Put the carrot and 1 cup water in a small saucepan. Bring to a simmer over medium-low heat and cook until tender, about 15 minutes. Reserve 1/2 cup cooking liquid, then drain the carrot.
Puree the carrot and reserved cooking liquid in a blender until smooth. Add the ginger, brown sugar, vinegar, lemon juice, soy sauce, sesame oil and teaspoon salt; pulse until smooth.
Photograph by Andrew McCaul
Courtesy of Food Network Magazine