In a medium bowl, whisk together the eggs, egg yolks, 1 cup of the parmesan cheese, salt, and pepper, until combined. Set aside.

Heat a large, deep skillet over medium-high heat and add the bacon. Cook for 8-10 minutes, until bacon is brown and crispy. Remove bacon from pan and let drain on a paper towel lined plate. Pour all bacon fat, leaving 1-2 tablespoons in the pan for the mushrooms, into a small bowl or ramekin to use for the sauce later. Set the reserved bacon fat aside. Meanwhile, bring a pot of water to a boil for the pasta and salt the water.

Add 1/4 cup olive oil to the skillet with the bacon fat. Heat over medium-high heat, and once hot, add the mushrooms along with some freshly cracked black pepper. Cook for 12-15 minutes, leaving the mushrooms mostly undisturbed (tossing them 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 in a bowl. Add the bacon fat to the skillet so that it's ready for the pasta!

At this point, your 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, toss the pasta, using tongs, to combine it with the mixture to create a creamy "sauce". Add the remaining 1/2 cup of parmesan and keep tossing until combined. Add a splash or two more of the pasta water if it seems like it needs more moisture. Season with freshly cracked black pepper (you likely won't need any salt since bacon and parmesan are salty enough on their own!) and fresh parsley and toss once more. Top with the crispy mushrooms. Serve immediately with extra grated parmesan and fresh basil leaves.