Bring a soup pot full of hot water to a boil. In a large saucepan over medium-high heat, saute leeks in oil until soft and slightly golden, about 8 minutes. Stir in stock, garlic, and capers. Reduce heat to low; simmer 5 minutes. Season with salt and pepper to taste.
Add linguini to boiling water; cook until al dente. Drain and add to leek mixture along with parsley and lemon zest. Toss well to mix; then season with additional salt and pepper. Portion linguini onto four pasta plates. Sprinkle with Parmigiano-Reggiano.