Combine 1-/2 Tbs. of the olive oil with the parsley, mint, olives, capers, sherry vinegar, lemon zest, cumin, sugar, and several grinds of pepper in a small bowl. Mix well and set aside.
In a 12-inch nonstick skillet, heat the remaining 3 Tbs. olive oil over medium-high heat. Add the carrots, turnips, and 1-1/2 tsp. salt. Toss well to coat the vegetables with the oil. (The pan will look crowded.)
Reduce the heat to medium, cover partially, and cook, stirring and flipping every 3 to 4 minutes with a spatula, for about 10 minutes. Add the shallots and 1/4 tsp. salt and stir well. Cover partially and continue to cook, stirring every 1 to 2 minutes, until the vegetables are tender and well-browned, 18 to 20 minutes. Reduce the heat if the vegetables brown too quickly.
Reduce the heat to medium low and add the kale, gently mixing until wilted. Stir in the parsley-mint sauce to taste, and remove the pan from the heat. Season to taste with more salt and pepper, and serve.
Pair any leftover sauce with grilled steak or chicken.