In a medium mixing bowl, whisk together the eggs, egg yolks, 1 cup of the parmesan cheese, salt, and pepper, until smooth. Set aside while you prepare the rest of the ingredients.

Heat a large skillet over medium-high heat and add the chopped bacon. Cook for 8 minutes, or until bacon is golden brown and crispy. Remove bacon from pan and add to a paper towel lined plate. Pour bacon fat into a small bowl, leaving 1-2 tablespoons in the pan for the mushrooms. Save the reserved bacon fat for the sauce later. Meanwhile, bring a pot of salted water to a boil for the pasta.

Add ¼ cup olive oil to the skillet with the 1-2 tablespoons of bacon fat. Heat over medium-high heat, and once hot, add the mushrooms and a few cracks of freshly ground black pepper. Cook for 12-15 minutes, tossing the mushrooms only a few times, so that they get deeply golden and crispy. Once they're finished, season with salt and stir in the chopped rosemary. Remove mushrooms from pan and set aside. Add the reserved bacon fat to the skillet so that it's ready for the pasta!

At this point, your pasta water should be boiling. Add your pasta and cook according to package directions, or until just al dente. Remember to reserve your pasta water (I just scoop a bunch out with a large mug or heat-safe glass measuring cup!) Drain pasta, and quickly pour it back into the pan with the bacon fat. Quickly add the egg-parmesan mixture and a splash of pasta water (start with 1/4 cup) and working quickly, using tongs, toss the pasta, to combine it with the mixture to create a creamy "sauce". Add the remaining 1/2 cup of parmesan cheese and keep tossing until melted and combined. Add a splash or so more of the pasta water (up to the remaining ¼ cup) if it seems like it needs more moisture. Season with freshly cracked black pepper. You most likely won't need any salt since bacon and parmesan are salty enough on their own! Add chopped fresh parsley and toss once more. Top with the crispy mushrooms. Serve immediately with extra grated parmesan and fresh basil.