Bring a large saucepan of salted water to a boil over high. Add pasta, and gently stir; cook until very al dente, about 2 minutes less than package directions. Meanwhile, bring chicken stock to a simmer in a large straight-sided skillet over medium-high. (If stock reaches a simmer before pasta is ready, reduce heat to low until ready to use.)
Just before pasta is ready, ladle 1 cup pasta cooking liquid into skillet with the stock. Transfer cooked pasta to skillet using kitchen tongs or a spider. (Reserve remaining pasta cooking liquid.) Cook pasta and stock mixture over medium-high, stirring and shaking skillet constantly, adding butter, 2 tablespoons at a time, until a creamy but very loose sauce forms that coats the pasta, 1 to 2 minutes.
Remove skillet from heat; gradually stir in cheese until melted and sauce is smooth, 1 to 2 minutes. If needed, gradually stir in reserved pasta cooking liquid, ¼ cup at a time, until sauce coats pasta but is still loose enough to pool in bottom of skillet. Season with salt to taste. Serve immediately.
