8 slices bacon, cut into 1/2-inch pieces
1/2 cup water
3 cloves garlic, minced
2 1/2 ounces Pecorino Romano cheese, grated (1 1/4 cups)
3 large eggs, PLUS
1 large yolk
1 teaspoon pepper
1 pound spaghetti
1 teaspoon salt
Bring bacon and water to simmer in 10-inch nonstick skillet over medium heat; cook until water evaporates and bacon begins to sizzle, about 8 minutes. Reduce heat to medium-low and continue to cook until fat renders and bacon browns, 5 to 8 minutes longer. Add garlic and cook, stirring constantly, until fragrant, about 30 seconds. Strain bacon mixture through fine-mesh strainer set in bowl. Set aside bacon mixture. Measure out 1 tablespoon fat and place in medium bowl. Whisk Pecorino, eggs and yolk, and pepper into fat until combined.
Meanwhile, bring 2 quarts water to boil in Dutch oven. Set colander in large bowl. Add spaghetti and salt to pot; cook, stirring frequently, until al dente. Drain spaghetti in colander set in bowl, reserving cooking water. Pour 1 cup cooking water into liquid measuring cup and discard remainder. Return spaghetti to now-empty bowl.
Slowly whisk 1/2 cup reserved cooking water into Pecorino mixture. Gradually pour Pecorino mixture over spaghetti, tossing to coat. Add bacon mixture and toss to combine. Let spaghetti rest, tossing frequently, until sauce has thickened slightly and coats spaghetti, 2 to 4 minutes, adjusting consistency with remaining reserved cooking water if needed. Serve immediately.