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Put the hot dogs into a hot grill pan and cook until golden brown, about 10 minutes. Cool the hot dogs to room temperature, then cut them crosswise into bite-size pieces.
Toss the potatoes, parsley, basil, and olive oil in a large bowl. Season the potato salad with salt and pepper, to taste. Add the hot dogs to the salad and toss. Season to taste, adding additional olive oil if the salad seems too dry. Serve the salad at room temperature.