Step 1

Bring a large pot of heavily salted water to a boil. Meanwhile, cut 8 ounces slab bacon into 1/2-inch cubes, or cut 8 ounces thick-cut bacon crosswise into 1/2-inch pieces. Place in a large nonstick or cast-iron skillet and cook over medium heat until fat is rendered and bacon is crispy, 9 to 10 minutes.

Step 2

Meanwhile, finely grate 1 ounce Parmesan cheese (about 1/2 cup firmly packed) or measure out 1/3 cup store-bought grated. Finely grate 1 ounce white sharp cheddar cheese (about 1/2 cup). Place both cheeses, 4 large egg yolks, 1 large egg, and 1/2 teaspoon black pepper in a medium bowl and whisk until combined.

Step 3

When the bacon is ready, transfer it to a small bowl with a slotted spoon and reserve the pan.

Step 4

Add 1 pound dried spaghetti or linguine to the boiling water and cook until al dente, about 8 minutes or according to the package instructions. Meanwhile, heat the bacon fat in the skillet over medium-high heat until shimmering. Cooking in 2 batches if needed, add the remaining 4 large eggs (or 6 if serving and fry until the whites are crispy and the yolks are still runny, 3 to 4 minutes. Tilt the skillet slightly while they cook so the bacon fat pools and use a large spoon to baste the whites with the fat as needed. Season with kosher salt and black pepper and transfer to a baking sheet or plate and keep warm near the stove.

Step 5

When the pasta is ready, reserve 1 cup of the pasta cooking water. Drain the pasta and return it to the pot over medium-low heat. Add the bacon and a splash of the reserved pasta water and toss to combine. Turn off the heat. Add the cheese mixture and stir with tongs, adding more pasta water as needed, until the sauce is thickened and coats the pasta, about 2 minutes. Taste and season with kosher salt and black pepper as needed.

Step 6

Divide the pasta between 4 or 6 bowls and top each with a fried egg. Serve immediately, as this does not reheat well.