Heat 2 tablespoons olive oil in a large heavy-bottomed skillet (preferably cast iron) over medium-high heat. Add the potatoes and cook, stirring often, until lightly browned, 7 to 8 minutes. Add the onion and cook, stirring, until softened, 2 to 3 minutes. Reduce the heat to medium. Add the remaining 2 tablespoons olive oil, the bell pepper and zucchini; cook, stirring, until just tender, 7 to 8 minutes.
Add the pork to the skillet and sprinkle with the chili powder. Cook, stirring occasionally, until the pork is heated through, about 2 minutes. Increase the heat to medium high and cook until the pork and vegetables are golden brown, 2 to 3 more minutes. Add the kale, cherry peppers and brine; toss until the kale is wilted, about 1 minute. Stir in the basil and season with salt. Remove from the heat and stir in the cheese curds.