Using a small bowl, whisk together the eggs, egg yolks, Parmesan, Pecorino and pepper; set aside. (Tip: Do this first to allow egg mixture to come to room temperature before using it later in the recipe.)

Place a large, deep skillet over medium-high heat and add olive and butter. Once butter has melted, add mushrooms. Cook for 13-15 minutes, stirring occasionally, or until mushrooms are golden brown.

Reduce heat to medium-low; add onions, garlic and salt. Continue cooking, stirring occasionally, for 3-4 minutes, or until onions have begun to soften. (If necessary, reduce heat to low to keep mushroom mixture warm until pasta is ready.)

Meanwhile, cook the pasta in salted water for 2 minutes less than package instructions. Reserve 2 cups of the pasta water; drain rest of pasta water into a large serving bowl. (See note below)

Transfer cooked pasta along with 1½ cups of the reserved pasta water into skillet with the mushrooms. (If heat was reduced, increase heat back to medium-low.)

Continue cooking for 2-3 minutes, stirring occasionally.

Pour water out of the large serving bowl. Transfer pasta and mushrooms into the bowl; stir in the egg mixture and chopped parsley. Stir constantly until eggs begin to thicken and coat the pasta. (The heat from the pasta and bowl will indirectly cook the eggs. If the mixture becomes too thick, add a splash of the reserved pasta water. I typically add ~¼ cup of additional pasta water at this point.)

Top liberally with freshly ground black pepper and additional Parmesan cheese. Serve immediately.