Preheat oven to 425°F. Toss together brussels sprouts, sweet potato, 1 tablespoon of the oil, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper in a large bowl. Arrange in an even layer on a large rimmed baking sheet. Bake in preheated oven until tender, about 20 minutes.
Meanwhile, stir together yogurt, water, parsley, chives, dill, lemon zest, lemon juice, tarragon, anchovy paste, garlic, 1 tablespoon of the oil, and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Set aside until ready to use.
Place kale and remaining 1/2 tablespoon oil in a bowl; massage gently with your hands until slightly tender, about 3 minutes. Arrange kale, chicken, and baked brussels sprouts mixture in a large bowl. Drizzle with yogurt dressing. Sprinkle with pepitas and cranberries.