Step 1

In a medium saucepan, combine the lentils, 2 cups of the water, and 3/4 teaspoon of the salt. Bring to a boil, reduce the heat, and simmer, partially covered, until the lentils are just tender but not falling apart, 25 to 30 minutes. Drain if necessary, cover, and set aside.

Step 2

Meanwhile, in a large nonstick frying pan or a wok, heat 1 tablespoon of the oil over moderately high heat. Add half of the ginger and scallions and cook, stirring, for 30 seconds. Add the turnips and carrots and cook, stirring, for 2 minutes. Add 1 cup of the water, 3/4 teaspoon of the salt, and the vinegar. Boil until the vegetables are tender and no liquid remains in the pan, about 10 minutes. Transfer the vegetables to a large bowl.

Step 3

In the same pan, melt the butter with the remaining 2 tablespoons of oil over moderately high heat. Add the remaining ginger and scallions and cook, stirring, for 30 seconds. Add the Brussels sprouts and cook, stirring, for 2 minutes. Add the green beans and the remaining 1/3 cup of water and 1 teaspoon salt. Reduce the heat and simmer until the vegetables are just tender, about 5 minutes. Stir in the carrot mixture and the drained lentils and toss gently to combine.