Heat the olive oil in a 4-quart heavy saucepan or Dutch oven over medium heat. Add the onions, carrots, and sweet potatoes with ¼ teaspoon salt and sauté until the onions are translucent and carrots and sweet potatoes and slightly soft, stirring occasionally, about 7 minutes. Then add the garlic, cumin, and allspice, stir, and cook for one more minute.
Add the lentils and stock and bring to a boil. Lower the heat to medium-low and simmer uncovered until the lentils are tender and have absorbed most of the liquid, 20 to 30 minutes.
Add extra stock if needed. Season with salt. Stir in the kale and cook until the kale wilts.
Garnish with cilantro and serve with the pita bread.