Thinly slice the scallions, separating the green and white parts. Mince the onion. Cut the sweet potato into ½-inch dice. Prepare the tomatoes by chopping roughly, saving the juice. Stem the kale and cut the kale leaves into ¼-inch strips.
Heat a stockpot or other tall wide pot over medium heat. Add the oil and swirl to coat the bottom. When the oil is hot, add the scallion whites, onions and sweet potatoes, season with salt and pepper, and sauté until the onions have browned lightly, about 5 minutes. Add the ginger, tomatoes with their juice, stock and tamari.
Place the miso in a small strainer, submerge it in the stockpot, and whisk it until it has dispersed into the liquid and remove the strainer. Adjust the seasoning, if necessary. Bring the stew to a simmer and cook until the potatoes are soft and the liquid is reduced by a quarter, about 10 minutes. Adjust the seasoning again, if necessary.
Add the kale and simmer until soft, 1-2 minutes. Taste the seasoning a final time and adjust, if necessary.
Toast and quarter the pitas. Transfer the stew to four individual soup bowls. Drizzle with the olive oil and serve with the pita.