How to Make It
Fill up a large pot with water to 2 inches from top of pot; generously salt water. Bring water to a boil over high. Stir in pasta, and return to a boil over high. Cook pasta, stirring occasionally, until very al dente (2 minutes less than package directions for al dente). Drain pasta in a colander, reserving 1 1/2 cups cooking liquid in a small heatproof bowl. Add a small drizzle of oil to pasta in colander, and toss gently to coat.
Cook oil and garlic in a large, deep skillet over medium, stirring often, until lightly toasted, about 4 minutes. Add capers and black pepper; cook, stirring often, 2 minutes. Add 1 cup reserved pasta cooking liquid; bring to a boil over medium. Stir in cooked pasta, and return to a boil over medium; cook until pasta is al dente, 3 to 4 minutes, adding up to 1/4 cup reserved pasta cooking liquid until desired sauce consistency is reached. Remove from heat.
While stirring pasta mixture vigorously using a wooden spoon, pour egg yolks in a quick stream into mixture; stir until slightly creamy, about 10 seconds. While stirring vigorously, quickly add cheese and chives; stir until sauce is creamy and silky, about 20 seconds. Gently stir in crab and lemon zest and juice. Season to taste with fine sea salt or kosher salt. (If needed, add additional pasta cooking liquid to loosen sauce.) Divide pasta mixture evenly among 4 shallow bowls; garnish with additional cheese. Serve with garlic bread and lemon wedges.