Bring a large pot of salted water to a boil over medium-high heat. Add 1 pound dried spaghetti and cook according to the package instructions until al dente, 8 to 12 minutes. Reserve 1 cup pasta water. Drain the spaghetti.

Meanwhile, cook 8 ounces diced pancetta in a large skillet over medium-low heat, stirring occasionally, until browned and crisp, 8 to 12 minutes. Stir in 3 tablespoons tomato paste and cook, stirring occasionally, until deep red and it sticks to the pan, 1 to 2 minutes.

Stir in 12 ounces halved cherry tomatoes and ¼ cup of the pasta cooking water. Cook, stirring regularly and scraping the bottom of the pan with a wooden spoon to release any browned bits, until the tomatoes are just softened, 2 to 3 minutes.

Use tongs to transfer the spaghetti to the skillet. Toss well to coat the pasta. Remove the skillet from the heat and cool slightly.

Meanwhile, whisk 4 room-temperature large eggs and 2 room-temperature large egg yolks together in a medium bowl. Whisk in 1 ½ ounces finely grated Pecorino Romano cheese, ½ teaspoon kosher salt, and ½ teaspoon black pepper. While whisking constantly, slowly stream in 2 tablespoons of the pasta water until well-combined.

While tossing the pasta constantly with tongs, slowly start pouring the egg mixture into the skillet, pouring over the noodles and not the edges of the pan. Add ¼ cup of the pasta water, return the pan to the lowest heat, and continue to cook, tossing constantly, until the sauce starts to thicken and coats the pasta, 1 to 2 minutes. (Add more pasta water a tablespoon at a time as needed for a thinner sauce.)

Remove the skillet from the heat. Taste and season with more kosher salt as needed. Serve immediately with more grated Pecorino and a few grinds of pepper on top if desired.