Scoop flour into a pile on a clean table. Form a well in the center using a fork.
Pour Parmesan, eggs, whole milk ricotta, part-skim ricotta, melted butter, salt, and pepper into the well.
Using the fork, slowly incorporate dry ingredients into wet ingredients, pulling the rim of the well into the center. Once dough begins to form a ball, knead it until the dough comes together and is evenly combined, 2 to 3 minutes.
Place dough ball on a large plate; cover with plastic wrap to prevent dough from drying out as you work with it.
Dust a clean surface with flour, then trim off about 1/4 of the dough. Roll this portion into a long round cylinder, about 1-inch in diameter. Slice this strip into 1-inch gnocchi. Repeat this process to shape and cut all of the dough into gnocchi.
Lightly flour a sheet pan. Roll each gnocchi over a microplane or the side of a box grater to add texture to the exterior. Place gnocchi on the sheet pan.
Bring a large pot of salted water to a boil. Add gnocchi, being careful not to overcrowd the pot. Bring water back to a boil and cook until gnocchi begins to float and the pieces have reached your desired texture, 3 to 5 minutes. Top with your favorite sauce and Parmesan.
