Set a large, 1-gallon stock pot with a pasta insert 3/4 full with water over high heat and bring to a boil. Add 1/4 cup of the salt to the water as it boils. Place the linguine in the salted water and return to a boil, stirring occasionally. Cook the pasta until nearly al dente, 10 to 12 minutes. Drain and set aside, reserving 1 cup of the pasta cooking water.

While the pasta cooks, set a 14-inch saute pan over medium-high heat and add the olive oil and butter to the pan and once the butter melts, add the onions to the saute pan. Cook the onions, stirring occasionally until they have softened and become caramelized, about 10 to 12 minutes. Add the garlic, olives and rosemary to the pan, stirring until fragrant, about 1 minute. Add the reserved pasta and reserved pasta cooking water to the pan and season with the remaining 2 1/2 teaspoons kosher salt, pepper, and parsley. Cook, tossing to combine, until the water has nearly evaporated, about 3 to 4 minutes. Remove from the heat, sprinkle with the grated cheese and toss to combine. Serve while hot.