Bring a large pot of water to a rolling boil. Add enough salt so the water tastes pleasantly seasoned. Drop in the spaghetti and cook until just al dente, stirring once or twice so the strands do not stick together. Before draining, scoop out at least ½ cup of the hot pasta water and set it aside.
While the pasta cooks, place a wide pan over medium heat and add the olive oil. Once warm, add the beef bacon or smoked beef strips. Cook slowly, stirring occasionally, until the pieces release their fat and turn evenly golden and crisp around the edges. This should take about 6–8 minutes.
Add the minced garlic to the pan and cook for 20–30 seconds, just until fragrant. Do not let it brown. Lower the heat to the gentlest setting and keep the pan warm.
In a mixing bowl, whisk together the egg yolks, whole egg, heavy cream, grated cheese, black pepper, and salt. Whisk until smooth and well combined. The mixture should look thick but pourable.
Drain the spaghetti and immediately add it to the warm pan with the beef bacon. Toss gently so the hot pasta picks up the rendered fat and flavor.
Remove the pan from direct heat. Pour the egg and cream mixture over the spaghetti. Using tongs, toss continuously so the sauce coats every strand. The residual heat from the pasta will thicken the sauce without scrambling the eggs.
Add the reserved hot pasta water, a little at a time, tossing between additions, until the sauce becomes glossy, creamy, and softly flowing. The finished sauce should cling to the pasta and pool lightly at the bottom of the pan.
Taste and adjust seasoning with extra black pepper or salt if needed. Serve immediately, finished with a light sprinkle of chopped parsley if desired.
