Bring a medium pot of water to a boil, add the pasta and cook according to package instructions; drain, reserving 1/2 cup of the pasta water. Return the pasta to the pot.
Meanwhile, heat the olive oil in a large nonstick skillet over medium-high heat. Add the soppressata and cook, stirring, until blistered and the oil turns reddish, about 3 minutes. Add the artichoke hearts and olives and stir to coat. Add the tomatoes, the reserved pasta water, basil leaves and 2 tablespoons of olive brine. Bring to a simmer and cook until most of the water evaporates and the sauce thickens slightly, 8 to 10 minutes.
Add the sauce and mozzarella to the pasta in the pot and stir to combine. Divide among 4 bowls and garnish with more basil. Serve either hot or cold.
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