First, bring a large pot of salted water to a boil.
Next, add jarred sun dried tomatoes, water, red wine vinegar, tomato paste, minced garlic, and salt to a blender. Blend on medium speed until smooth, about 30 seconds to 1 minute.
Transfer the sauce to a large sauté pan and warm over medium heat until lightly steaming, about 2 to 3 minutes. Crumble three-quarters of the goat cheese into the sauce and stir to incorporate as the cheese melts, about 3 to 4 minutes. Then turn heat to low to keep warm.
While the sauce warms, boil the potato gnocchi until a few begin to float to the top of the pot. Use a spider strainer or slotted spoon to scoop them out of the pot, and check for doneness by gently pressing into the gnocchi with a fork (if they are soft, they are done cooking). Transfer the cooked gnocchi to the sauce.
Stir to combine gnocchi with the goat cheese tomato sauce, then remove from heat. Plate with a garnish of crumbled goat cheese and a fresh sprig of basil, and serve immediately.